discrete coded waveforms for signal ...ir.amu.ac.in/2456/1/t 5181.pdfcontents chapter 1...
TRANSCRIPT
DISCRETE CODED WAVEFORMS FOR SIGNAL PROCESSING IN RADAR
SYNOPSIS
T H E S I S SUBMITTED FOR THE AWARD OF THE DEGREE OF
fiottor of $I|tlos;opIip IN
ELECTllONICS ENGINEERING
BY
ZIA AHMAD ABBASI
UNDEH THE SUPERVISION OF
Dll. PARID GHANI
DEPARTMENT OF ELECTRONICS ENGINEERING ALIGARH MUSLIM UNIVERSITY
ALIGARH (INOfA)
pTttAO t/ft, 4«v
SYNOPSIS
The work presented in this thesis is concerned with the design of
discrete coded wavefonns for improving range resolution and clutter
performance of radar systems. This approach to signal design offers many
advantages in terms of waveform shaping and digital implementation of
processors. Assuming a matched filter receiver the bulk of the work is
concentrated on studying the autocorrelation function properties of these
waveforms, which are directly related to the range resolution. Pulse trams of
this type are also useful in synchronising digital communication systems.
Sequences of this type can also be used as an orthogonal code in a simple
coding system.
There is a broad class of discrete coding waveforms or sequences,
that can be found with good autocorrelation function. The class of binary
sequences is important. These are easy to generate and store using
standard logic circuits, cross-correlation is simpler and transmitters operate
continuously at peak power. The binary sequences having values ±1 such as
Barker sequences with good autocorrelation fimction, have the most
uniform distribution of energy for the given length. However, Barker
sequences with a length greater than thirteen do not exist. Furthermore, the
autocorrelation function of binary sequences contains significant sidelobes
as compared to the non-binary sequences. The non-uniform or multilevel
sequences such as Hufifinan sequence have got autocorrelation function
much similar to a pulse. But the cost for this achievement is paid in the form
of circuit complexity which is due to the need of amplitude and phase
modulation hi its implementation. However, the use of digital microchcuits
becoming more attractive in signal processing applications due to their
flexibility, cheapness and compactness, the generation of multilevel
sequences are no longer difficult. Few disadvantages are also associated with
multilevel sequences. The distribution of energy is not uniform throughout
the sequence length, the energy efficiency of the multilevel sequences are
poor as compared to the binary sequences.
An attempt is made to solve the signal design problem usmg
numerical optimisation methods that incorporate the fixed amplitude
constraint. In particular, a constraint optimisation technique is developed
for synthesising binary sequences with good autocorrelation function
properties. It is well known that by choosing a large number of elements
randomly, sequences where r.m.s. sidelobe levels are of the order /N can be
found. However, it can be expected that in the statistical synthesis a large
number of sidelobes will exceed /N. It will be shown that by proper
choice of the sequence both average and peak sidelobe can be held at a
lower value and this yield a better range resolution and clutter rejection.
Moreover, the problem of designing pairs and sets of phase coded
pulse trains with low autocorrelation sidelobes and small mutual cross-
correlation is considered. Such sequences have many practical application.
For example they may be used as address codes in a tune division multiple
access (TDMA) system, where information from several data sources is to
be transmitted over a channel.
In the case of non-binary sequences such as HufiBnan codes
(impulse equivalent sequences) a synthesis method based on the weighting
sequences of the optimum inverse (least square energy) filters is presented
which yields sequences with good energy efficiency and near pulse like
autocorrelation fimction. It is shown that the tap gains or weighting
coefficients of the optimum inverse filter has better autocorrelation function
than the input sequence itself This fact is being utilized to generate the
sequence of exfremely small autocorrelation function sidelobes by using the
process of inversion a number of times.
Although, the range sidelobes can be reduced quite effectively by
proper signal design methods, for some applications they might still be too
large. Thus sidelobe reduction methods, which minimise the detection loss
subject to a set of sidelobe constraints, are presented by mismatching the
receiver filter.
An artificial neural network (ANN) based approach is also
developed which can completely reduce the autocorrelation sidelobes down
to zero value at the output of the matched filter. Various possibilities are
explored and their behaviour in the presence of noise is considered. ANN
has found wide applications in pattern recognition. The pattern recognition
is also applicable in radar signal detection or classification processes. In the
present case radar signals can be treated as data patterns. The basic theories
of statistical hypothesis testing, decision theory etc. apply in this situation.
Indeed, many pattern recognition techniques employ likelihood ratios, when
the statistics are known apriori, and pattern recognition or classification
is then merely an applications of multiple hypothesis testmg. Nevertheless,
some interesting techniques presented under the name of pattern
recognition, learning machines, artificial intelligence etc. are applicable to
radar system and should be available for the radar designers consideration.
Finally, in the case where significant target velocity is encountered it
becomes necessary to consider not only the range but also the velocity
resolution (Doppler shift) properties of the transmitted waveform. Combined
range and velocity resolution depends on the complete waveform structure in
time and fi-equency. Hence signals with good resolution in one parameter
may perform very poorly when combined resolution in both parameters
is considered. For combined resolution in range and velocity the wave
form must be investigated in terms of the complete MF response in delay
and Doppler. This generalised response for various types of pulse trains is
considered by using the standard range Doppler ambiguity description
^
(ABF) of Woodward. The ABF plays a central part in the analysis of
combined resolution. This is so because the width of the main response
peak of the ABF serves as a measure for close-target visibility in range-
Doppler, while the low level response and subsidiary spikes give an
indication of the self-clutter and target masking problem by mutual
interference.
DISCRETE CODED WAVEFORMS FOR SIGNAL PROCESSING IN RADAR
^ T H E S I S SUBMITTED FOR THE AWARD OF THE DEGREE OF
IBottor of $I)ilo)iopIip
ELECTRONICS ENGINEERING
•» ' '
> •^••<j v *
BY . ^
ZIA AHMAD ABBASI
.*^» !»••
UNDER THE SUPERVISION OF
DR. FARID GHANI
DEPARTMENT OF ELECTRONICS ENGINEERING ALIGARH MUSLIM UNIVERSITY
ALIGARH (INDIA)
1 9 9 7
T5181
2 5 DEC 1999
i^
CONTENTS
CHAPTER 1 cINTRODUCTION
1 1 Background 1
12 Outiine of Investigation 5
CHAPTER 2 : SIGNAL PROCESSING CONCEPTS AND W A V E F O R M DESIGN
2 1 Introduction 8
2 2 Representation of Pulse Train 9
2 2 1 Complex Envelope and Bandpass Filtenng 10
2 2 2 The Z-Transfonn 14
2 3 Optimum Processing of Radar Signals 18
23 1 Digital Matched Filters 20
2 3 2 Pulse Compression Radar 27
2 3 3 Ran^e Resolution m a M F Radar 28
CHAPTER 3 :SYNTHESIS O F PULSE TRAINS FOR SPECIFIED AUTOCORRELATION FUNCTION
3 1 Introduction 34
3 2 Synthesis of Pulse Trains if the ACF IS known in Magmtude and Phase 35
3 3 Synthesis of Pulse Trains if only the Magnitude of ACF is known 38
3 4 Discrete Phase Approvunation to Linear FM Signals 41
3 41 Properties of the Compressed Pulse Trams 44
CHAPTER 4 :DESIGN OF BINARY CODED PULSE TRAINS
4 1 Introduction 48
4 2 The Optimization Problem 49
4 2 1 Formulation of the Synthesis Problem 50
4 2 2 Synthesis Using Element Complementation 53
4 2 3 Synthesis Using Cyclic Shift and Bit Addition 55
4 3 Uniform Sequences with Low Auto-correlation Sidelobes and Small Crosscorrelation 56
43 1 Statement of tlie Problem 57
4 3 2 Results of the Synthesis 60
4 4 Ternary Sequences 62
4 5 Summary 64
CHAPTER 5 :AMPLITUDE AND PHASE MODULATED PULSE TRAINS 5 1 Introduction 67
5 2 Hulliiiiiii Sequences 67
5 3 Design Based on Optimization 69
5 4 Clipping Method 71
5 5 Iterative Process for tlie Generation of Multilevel Sequences 73
5 5 7 Introduction 73
5 5 2 Optimum Inverse (Least Error Energy) Filter 74
5 5 3 Results and Discussion 76
5 6 Choice of Slnrdng Sequence 79
5 6] Choice At Random 79
5 62 Effect ofOptimumPulse Position 79
5 6 3 Conditions for Convergence 80
5 7 Averaging the Spiking Filter Coefficients with Input Code 81
5 8 Summary 84
CHAPTER 6 :SU)ELOBE REDUCTION 6 1 Introduction 86
6 2 Optimum Inverse Filler 88
621 An Iterative Method for Range Side Lobe Suppression for Binary Codes 91
6 3 A Simple Scheme for Side Lobe Elimination 92
6 4 Neural Network Approach for Side Lobe Suppression 94
6 5 Summary 99
CHAPTER 7 :COMBINED RANGE AND RANGE RATE RESOLUTION 7 1 Introduction 101
7 2 Ambiguity Function of Pulse Trains 102
CHAPTER 8 :CONCLUSION AND DISCUSSION 105
APPENDICES A Recursive Solution of Simultaneous Equations Involving an Auto-correlation Matrix 109
B List of Sequences 117
REFERENCES 123
ABSTRACT
The work presented in this thesis is concerned with the design of discrete
coded waveforms for improving range resolution and clutter performance of radar
systems. This approach to signal design offers many advantages in terms of
waveform shaping and digital implementation of processors. Assuming a matched
filter receiver the bulk of the work is concentrated on studying the autocorrelation
function properties of these waveforms, which are directly related to the range
resolution. Pulse trains fo this type are also meful in synchronizing digital
communication system. Sequences of this type can also be used as an orthogonal
code in a simple coding system.
There is a broad class of discrete coding waveform or sequences, that can
be found with good autocorrelation function. The class of binary sequences is
important. These are easy to generate and store using standard logic circuits,
cross-correlation is simpler and transmitters operate continuously at peak power
due to uniform distribution of energy. However, the auto-correlation function of
binary sequences contains significant sidelobes as compared to the non-binary
sequences. The non-binary or multilevel sequences such as Huffinan sequence
have got auto-correlation function much similar to a pulse, but the distribution of
energy is not uniform throughout the sequence length, therefore, the energy
efficiency of the multilevel sequences are poor as compared to the binary
sequences.
An attempt is made tofsolve the signal design problem using numerical
optimization methods. It is well known that by choosing a large number of
elements randomly, sequences whose r.m.s. sidelobe levels are of the order ^fW
can be found. Howevei', it can be expected that in the statistical synthesis a large
number of sidelobes will exceed JW. It will be shown that by proper choice of
the sequence both average and peak sidelobes can be held at a lowr value and this
yield a better range resolution and clutter rejection. c •
Moreover, the problem of designing pairs and sets of phase coded pulse
trains with low auto-correlation side lobes and small mutual cross correlation is
considered. Such sequences have many practical applications. For example they
may be used as address codes in a time division multiple access (TDMA) systems,
where information from several data sources is to be transmitted over a channel.
hi the case of non-binary sequences such as Huf&nan codes a synthesis
method based on the weighting sequences of the optimum inverse filters is
presented which yields sequences with good energy efiBciency and near pulse like
autocorrelation function.
Although, the range sidelobes can be reduced quite effectively by proper
signal design methods, for some applications they might still be too large. Thus
sidelobe reduction methods, which minimize the detection loss are presented by
mismatching the receiver filter.
An artificial neural network (ANN) based approach is also developed
which can completely reduce the autocorrelation sidelobes down to zero value at
the output of the matched filter. Various possibilities are explored and their
behaviour in the presence of noise is considered.
Finally, in the case where significant target velocity is encountered it
becomes necessary to consider not only the range but also the velocity resolution
(Doppler shift) properties of the transmitted waveofim. For combined resolution
in range and velocity the waveform must be investigated m terms of the complete
MF response in delay and doppler. This is done by using the standard range
doppler ambiguity description (ABF) of wood ward.
ACKNOWLEDGEMENT
The author would like to express his gratitude to his supervisor Prof. F.
Ghani whose genuine interest, valuable suggestions and encouragement made this
work possible. Thanks are also due to the Chairman, Department of Electronics
Engg. for the facilities provided. Thanks are also due to Mr. Akmal for typing the
report.
( Zia A. Abbasi)
CERTIFICATE
This is to certify that the Ph.D. Thesis entitled Discrete Coded Waveforms
for Signal Processing in Radar which is being submitted by Mr. Zia
Ahmad Abbasi, Reader in tlie Dept. of Electronics Engg., A.M.U. Aligarh
for the award of the Degree of "Doctor of Philosophy" of the university, is
a record of candidate's own work carried out by him under my
supervision. The matter in this Thesis has not been submitted for the
award of any other Degree or Diploma.
Dated:June 24,1997 Prof. Farid Ghani Dept. of Electronics Engg.,
Aligarh Muslim University, Aligarh
CHAPTER - 1
INTRODUCTION
1.1 BACKGROUND
Radar is a technique for remote sensing using radio waves. Its basic purpose is
to detect the presence of a target of interest and to provide information concerning
the targets's location, motion, size and other parameters. The problem of target
detection is accomplished in a typical radar system (Fig. 1.1) by transmitting a radio
signal and detecting the waveform reflected by the target in the presence of unavoidable
system noise and reflections from undesired scatterers (clutter). It a return signal of
adequate strength is received, it is further analysed to determine the target's range,
velocity and so forth. This process is known as parameter estimation. The range of the
target is determined by measuring the delay of the return signal. Similarly, the velocity
of the target can be estimated, neglecting higher order effects, by measuring the
shift in carrier frequency (doppler shift) of the received waveform. Furthermore, the
transmitted signal can be carefully chosen and generated so as to optimize its capability
for extracting theTarget detection and parameter estimation are difficult practical
problems, particularly for small targets at great distances. In principle, however, both
problems are simple when only a single target is present. Target resolution, which may be
defined as the capability of a radar system to recognise particular target in the
presence of others, is one of the most important but also most demanding tasks[l] & [2].
For high performance radar systems .these tasks become increasingly
complex. This explains the continuing effort being directed towards improving the
resolution capabilities of modern radars. Some improvements are still being made in the
components which affect radar performance, for example, receivers with low noise
figures, transmitters with higher output power, and antennas with more gain. Further
improvement can be obtained by means of elaborate signal processing schemes. In recent
years a considerable amount of work has been done in digital processing for radar The
Scatteres
Transmilter
Recehrer
Thertnal noise
Target (delay .doppler v)
Fig. 1.1 A typlcsA radar system
technical problems imposed by modern radar systems are those of processing (real time) a
large number of data and the requirement of complex signal processing operations.
These problems can only be solved reliably by the use of digital techniques, although
in some cases modern optical processing schemes can offer an alternative.
The choice of a suitable transmit waveform is an important problem in radar
design. This is so because the waveform controls resolution and clutter performance and
also bears heavily on the system cost. As compactness, cheapness and computational
speed of digital microcircuits continue to increase their use in signal processing
applications becomes more practical. In particular, the advent of solid-state antenna
arrays has its impact on radar system designers in two principal ways. First, peak
power limitations of solid state array elements have necessitated the use of waveforms
with long durations in order- to achieve the required signal energy over a desired
range. The required stability and reproducibility of such signals can only be satisfied
reliably by digital signal generation and processing. Secondly, the ability to switch the
beam of solid-state array at high speeds gives the radar a multi-function capability, thus
requiring the flexibility to enable a variety of waveforms to be employed, [3]. These
requirements have made digital signal processing with its inherent adaptability an
attractive alternative to analogue processing Theoretical studies which provide the
basis for technical advances have not, so far, solved the general signal design
problem. The knowledge of the properties of the pulse trains, a class of signals
particularly well suited to digital processing, i therefore, of increasing practical
importance.
An early suggestion for using discrete coded waveforms in radar appeared in a
paper by Siebert, [4] treating the general problems of radar. Siebert noted that
certain binary coded waveforms offered substantial improvement in range and velocity
resolution. However, it was shown that in order to obtain these improvements, it would
be necessary to employ long periodic binary sequences known as pseudo-random
sequences, [5]. Later, Lerner6 suggested that the periodic sequence could be modified to
form an aperiodic signal and yet retain the nearly optimum resolution property of the
waveform. The use of aperiodic signals allows the construction of passive matched filter
receivers. At about this time the signal design problem was approached in a slightly
different way. Assuming a matched filter receiver, the range resolution capability (in the
absence of doppler shift) was found to be directly related to the autocorrelation function
of the transmitted waveform. Therefore, the approach consisted of attempting to design
aperiodic binary sequences having optimum autocorrelation properties, [7] & [9].
These sequences were called 'Optimum finite code groups' or Barker sequences.
Since these early evaluation a number of authors have made valuable
contributions in the field of waveform design, [10], [11], ...[15]. An interesting analytical
method for generating binary codes was reported by Boehmer, [10] using number
theory. Another quite different approach to the problem is discussed in a paper by
Vekmanll and Varakin, [12]. The authors suggest a synthesis procedure on the basis of
spectral theory and the method of stationary phase.
Heimiller, [16], Frenk, [17] and Zadoff, [18] have shown that there are other
suitable codes if the restriction of 0°-180° phase shifting is removed[19]. In the case of
Frank codes[17], higher order poly-phase coded words can be generated by coding each
sub-phase into one of M. phases. Huffrnan[20] considered the problem of designing
amplitude and phase modulated pulse trains. He has shown that finite length signals with
nearly ideal autocorrelation function can be generated. This property, however, is
achieved at the expense of amplitude modulation which results in increased system
complexity and lower energy utilization at the transmitter. Neverthless, the additional
expense of encoding and decoding of amplitude and/or phase modulated waveforms may
be justified for radars that must cope with land clutter or operate in a dense-target
environment. However, the use of amplitude modulated pulse trains is precluded in most
high power applications due to the inevitable loss in energy efficiency.
In spite of considerable effort that has been devoted to the problem of designing
waveforms with high range resolution there seems to be a lack of signal design
techniques and theory. All present methods tend to contain an element of trial and
error and, moreover, rely on the skill and ingenuity of the designer. In short, the study of
the properties of pulse trains does not appear to have progressed much beyond an
understanding of the types described above. The currently accepted belief that there is
no ideal waveform is not surprising considering the various different tasks modern radar
systems have to perform. On the other hand, the inability to find an ideal waveform is
not an excuse for failure to search for locally optimum waveforms for specific radar
applications and environments.
The effort in this thesis is directed towards the improvement of a factor
which constitutes a fiandamental limitation to radar performance, namely the
transmitted waveform Although the ways in which the transmitted signal affects the
system performance are well understood, [21], there seems to be no obvious solution
to the problem of designing energy efficient pulse trains for high resolution radars
Therefore, the work presented in this thesis is concerned primarily with the study and
development of design methods for improving the range resolution capability of
pulse trains. The pulse sequences discussed later, besides representing an interesting
mathematical area, are also of practical significance in related fields such as digital
communication and navigation
1.2 OUTLINE OF INVESTIGATION
This thesis is concerned with a number of different aspects of the pulse train
design problem for radar systems As a basis for later work, the principles of waveform
and processor design are outlined in Chapter 2 This is carried out assuming matched
filtciing and digital signal gcnciatiun and piuccssing
In Chapter 3, the important problem of synthesizing pulse trains from
autocorrelation functions, which arc specified at discrete points in phase amplitude or in
magnitude only, is tackled The resolution properties of pulse trains approximating F M
type signals are also studied in this chapter
In Chapter 4, the signal design problem is approached via numerical optimization
techniques This is done by minimizing an appropriate defined performance measure
reflecting the resolution capability of a signal The optimization method was then applied
to generate the phase modulated pulse trains of various length This chapter also treats
a new method of designing binary sequences using cyclic shifting and bit addition.
Moreover, the properties of pairs of phase coded sequences having low auto
correlation sidelobes and small mutual cross-correlation are studied.
Much of the material included deals with purely phase coded sequences. For
certain applications, however, the use of amplitude modulation can provide a useful
means of improving range resolution and clutter rejection.
Chapter 5 presents the results of the synthesis of energy efficient amplitude and
phase modulated pulse trains. In connection with Huffman Codes a new approach to the
signal generation using cHpping is developed.
Chapter 6 stands apart somewhat from the other chapters in that it is concerned
with sidelobe reduction techniques using mismatched filters. The central problem in
mismatched filtering is to consider the trade-off possibilities between resolution and
degradation in signal detractability. A neural network scheme is also presented in this
chapter which can suppress the sidelobes to any desired level.
Finally, Chapter 7 considers the combined range and velocity resolution properties
of the pulse trains. This is done using the standard range-doppler ambiguity fiinction of
Woodward, [21].
CHAPTER -
SIGNAL PROCESSING CONCEPTS
AND WAVEFORM DESIGN
2.1 INTRODUCTION
In this chapter some of the general principles of waveform and processor design
are discussed. (The selection of the desired transmitted waveform and of the receiver
choice involves in general two separable design problems) The design problem in
general requires two criterion, first; the choice of transmitted waveform; secondly the
design of the receiver. The waveform chosen, must be such that it optimizes the
performance in the total environmental conditions such as clutter, dense target
environment etc. Unfortunately, no such waveform is available which will suit for any
environmental condition. Since there are a number of different ways to implement a near
ideal receiver with some given waveform, therefore the design of Radar processing
scheme (hardware) is generally treated as a different problem. Complexity and
reliability of the hardware scheme and the corresponding cost involved are the usual
criterion for the processor design.
The design of signal processing system becomes more complex if the number of
input and output channels are large in number. For modern high resolution radars the
complexity of the signal processing system and the amount of data to be handled easily
reaches critical limits. However, in this thesis, the study is confined to the coded
waveforms as modulating fiinctions. The various modulation techniques of modulating
the carrier with such function and the methods of implementing the processors will not be
considered here.
In this chapter, the problem of waveform design based on the standard range
Doppler ambiguity fiinction described by Woodward, [21] is also considered. The
discussion on ambiguity function is confined to the results achieved by some specific
waveform. The amount of the work only depends upon the study of the zero Doppler
cross-section of the ambiguity fijnction, because the main factor here, is the range
resolution capability of a signal. In the references, [22], [23], ...[26], the properties of
various classes of waveforms and their ambiguity fiinctions are given in details. The
general description of the ambiguity fijnction is, despite their wide study, not without
its limitations. While with modern computers it is not difficult to derive the ambiguity
1 rect(trO
-T/2
-T T t
Fig. 2.1 Rect ana tri nmctloiis
i i i i i i i i
- N T 1 4-
Fig. 2.2 AmpUtadt and phase modnlattd piiHit train
function for a particular waveform it is generally not possible to derive a specific
waveform starting with a given ambiguity function. It is also necessary to have a
complete knowledge of target and clutter environment before selecting an appropriate
waveform.
2.2 REPRESENTATION OF PULSE TRAINS
An important class of signals well suited to digital processing are the pulse
trains. In general these pulse trains are the waveforms expressed as a finite number of
contiguous. Coherent carrier pulses (sub-pulses) each modulated in amplitude, phase and
frequency. Analytically this class of signals can be expressed in the form
u(i) = f^a{nyec(i^-n)Cos{27t[f^+fin)]t + <^in)} (2.1)
n = 0 • '
Where
T = Pulse duration
fc = Carrier Frequency
a(n) = amplitude of the nth pulse
(|)(n) = Phase of the nth pulse
f(n) = Frequency deviation of nth pulse
and rect(t/T) is the rect. function shown in Fig. 2.1 and defined as
/ 1, / < i - ) = i T 0 elsewhere
reclO = \: I' \ (2.2)
It can be noted that for a given carrier frequency, fc, and sub pulse duration, T,
the signal is completely described by the ordered sequences {a(n)}, {<t)(n)}, {f(n)}.
Depending upon these sequences the, pulse trains can be easily divided into three
following categories.
o • •t
pi I-* • •a
•a lO
tJ
I g
O 9
n
1 if
Categories
Group I : {f(n)} = 0
Group II : {(|)(n)} = {f(n)} = 0, {a(n)} = 1.0
Group III: {a(n)} = 1.0, {(|)(n)} = 0
The sub class of pulse trains considered here belong to Group 1 and are
referred to as amplitude and phase modulated (am ph m) pulse trains, [27] and are
shown in Fig 2 2
In digital signal generation the generator presents a sampled and quantized
signal which can be used to modulate the given carrier either in amplitude Phase or
in frequency Therefore, a digital signal generator in general may be assumed as a time
clocked unit, in which the output signal can be reconstructed by filtering or
modulation or both, using the sequences of digital data produced by signal generator In
most radar application phase modulation, and particularly digital phase modulation, is
the most attractive modulation method In the case of phase modulation, the signal can be
reconstructed by either of the two methods shown in Fig 2 3 The digital phase sample
generator produces samples as binary numbers which are converted into Cos(|)(n) and
Sin(j)(n) and thus into samples of Sr(t) and S,(t) (Eqn 2 5) Consequently the bmary
number can be used to digitally phase modulate a carrier signal in a digital phase
modulator
2.2.1 Complex Envelope and Band Pass Filtering
The complex envelope concept simplifies the theory of band pass signals Since
am ph m pulse trains are band pass signals, therefore, can be explained using
complex envelope representation The band pass signal given in equation 2 1 can be
expressed as
u(t) = Re [S(t) exp 027tfct)] (2 3)
10
Where S(t) is the complex envelope of u(t) and is given by
Sit) = f^a{n)rect{~n)expiMn)) (2.4) n = 0 •*
By comparing the Equations (2.4) with Eqn. (2.1), it is clear that the absolute
value of S(t) is nothing but real signal envelope. Thus the band pass signal u(t) is
completely described by the knowledge of its carrier frequency, fc, and its low
frequency complex envelope S(t). In terms of real and imaginary parts of S(t), the band
pass signal u(t) is given by
u(t) = SXt) Cos(27tfct) - Si(t) Sin(27ifct) (2.5)
Where
S(t) = SXt) +jSi(t)
The low pass signals Sr(t) and Si(t) are called the in-phase and quadrature
components, respectively, of the band pass signal. From above it is clear that the
complex envelope S(t) is independent of the carrier frequency, fc. Therefore, it is
sufficient to consider the complex envelope as the transmitted signal and to ignore the
carrier term exp (jlizfct). The great advantage of complex signal representation is that,
the operation such as linear band pass filtering (Convolution) can be expressed
directly in terms of the complex envelope. In other words, bandpass filtering of a signal
can be treated simply in terms of complex lowpass signals, [27] & [28]. This filtering
operation and how it can be implemented in terms of real lowpass filters is shown in Fig.
2.4. Consider the discrete coded waveform consisting of (N+1) pulses representing the
impulse response of a linear filter as an ordered set of complex numbers
{h(n)} = (h(0),h(l),h(2) ... h(N)) (2.6)
Such sequence is often called as a time series Alternatively the sequence {h(n)}
with values h(n) = h(nT) can also be visualized as being generated by sampling the
complex envelope h(t) of the corresponding continuous waveform every T seconds The
magnitude of the complex number h(n) represents the amplitude of the n* pulse while
11
Input UjCt)
coa^lex envelope
Bandpass f i l t e r h(t)
coinplex envelope
Output UjU) - Uj^(t) * h(t)
8 , (t) - «» », (t) * Y tt)
DjCf) - 8(f) Oj^(f)
SjCf) - S r(f) Sj^(f)
• in 2* f t e<iuivalent lowpasa f i l t e r
s in 2vf t c
Fig. 2.4 Baseband f i l ter ing .
the angle of h(n) specifies its phase The complex envelope of the filter impulse
response g(t) can easily be obtained by convolving the time series of Eq (2 6) with
Woodward's rect function
git) = f^h{n)d(t-uT)®recti^)
g{t) = f^h(fi)] d{t-nT)rect{t-^dT n=0 _«, ' -'
g{t)^f^h{ti)rect{^-n) (2 7)
Where ® indicates the Convolution Operation
The Fourier Transform of Eq (2 7) can be found by using the familiar rules of
transform theory
S(t-T) o S(f) exp(-j27ifr)
rect(t/T) <^ T Sinc(fr)
Where
Sinc(fr) = Sin(7cfr)/7tfr
The spectrum of the complex envelope g(t) is therefore,
g{f) = 7Sincifnf,Kn)expi-j2nfm^ n=0
or
g(f) = TSinc(fr)H(f) (2 8)
It can be seen that the spectrum H(f) is periodic with a period of 1/T and it
represents the spectrum of the coded sequence h(n) as shown in Fig 2 5 The
relationship between the Fourier Transforms of an analogue signal and its sampled
version is thus given by
12
-w/2 0 w/2
Fig. 2.5 EITcct of time domain sampling on bandlimlted signal.
^(/) = S^.(/+f) (2 9)
Where
HAf)=]Kt)exp(-j2n/i)dt - 0 0
H(J)=j;^hinDcxp{-j2pJhT) n=-cD
The above relationship essentially formulates the time domain sampling theorem
which states that a continuous function of time whose spectrum is limited to the band
(±W/2) is completely defined by time domain samples taken at intervals of lAV
Similarly the input signal, having (M+1) pulses is represented by a(n) The
complex envelope e(t) of the output signal has a FT which is, according to standard
transform rules of linear filtering, the product of the FT's of the complex envelopes of
the input waveform and the filter impulse response scaled by a factor 'A
M N
E(f) = yAZ''(")^M-j2nftjT}(^h(n)exp(-j27tfriT)}rSmc\fr)
M N
n o t I)
E{f) = y2{J^c{w)txp{-jlnfmT}T'Swc\fr) (2 10) m-O
where
c(A:) = 2a(«) / i (^-«) , k=0, 1,2, , (M+N) n-O
The complex envelope of the output signal e(t) is given by the inverse FT(IFT) of
Eqn (2 10) Use of the relationship
T^Sinc^fT) <^Ttri(t/T)
13
leads to
e{t) = ^'j^c{^m)tri{^-m) (2 11) ^ m=0
The function tri(t/T) is shown in Fig 2 1 and is defined as
^ 1 - ^ \t\<T //-/(f) = r ^
[0 ,elsewhere
At non-integer multiples of the sub pulse duration, T, the complex envelope of
the output waveform is given by linear interpolation between adjacent values The
output number sequence c(m) thus specifies, except for a constant scale factor, the
output waveform at regular sampling instants T
The main conclusion fi-om the foregoing is that complex envelope
representation of pulse trains simplifies the discrete linear filtering process which can
be regarded merely as multiplication of polynomials In addition only the spectrum of
the coded sequences need be considered to specify the waveform at integer multiples of T
2.2.2 The Z-Transform
It has been shown in the previous section that the FT can be used to describe the
frequency properties of pulse trains Another compact notation for the FT of such
signals is the Z-transform (ZT), [31] The ZT is also a very convenient method of
representing a signal by a set of poles and zeros in the complex Z-plane This is quite
similar to Laplace transform techniques used for analogue systems which can be
represented by poles and Zeros in the complex S-plane
The z-transform of an arbitrary numbers sequence, a(n), is simply a polynomial in
powers of Z"', and can be given by
A(Z) = a(0) + a(l)Z-' + + a(N)Z-''
N A(Z)= Za(n)Z~" (2 13)
/i = 0
14
Where Z is usually expressed in the polar form Z = exp(ST). In general the
frequency variable Z has both real and imaginary parts. Thus if,
S = a+j27cf
Z = exp(a + j27:f)T = exp(oT) [Cos(27ifr) + jSin(27ifr)]
The variable Z is often referred to as a "shift operator", since exp(-j27cfr)
implies a time delay of T seconds while exp(j27ifr) represents a time advance of T
seconds.
If the ZT, A(Z), is evaluated on the unit circle in the Z plane (|Z| = 1), the
spectrum of the time series a(n) is obtained.
AiZ) N
1 n=0
z M = ^ ( / ) = I « ( « ) e x p ( - y 2 ; ? / > i r ) (2 14)
It is noted that the spectrum A(f) is a continuous function in frequency. In
practice, however, the spectrum of discrete time series, is usually evaluated using digital
computers. This means that the spectrum A(f) can only be estimated at discrete points in
f which is generally referred to as discrete Fourier transforms (DFT). Although the
spectrum can only be estimated at suitably chosen intervals in f, it can be shown that
for time limited or periodic signals such a discrete representation of the underlying
continuous function does not result in any loss of essential information, [31]. This is
sometimes referred to as the frequency sampling theorem
Using DFT Equation (2.14) can be now rewritten as a finite-length sequence.
AiK)=tain)expi^) (2.15)
Where k = 0, 1,2, , N
Thus the frequency spacing between successive harmonics is 1/(N+1)T and the
frequency of the kth harmonic is therefore k/(N+l)T. It can be seen from equation
(2.15) that the sequence A(k) is periodic with a period of (N+1), i.e.
15
A(a) = A(N+1) A( 1) = A(N+2) , etc
Similarly the inverse transform (IDFT) of Equation (2 15) can be written as
\iy + 1 ) *=o
where n = 0,1,2, N
Where the multiplying factor 1/(N+1) has been included for convenience
Alternatively the above equations can be expressed in matrix form
G. — (N + l) ' 4 (2 17)
Where the (N+1) element cohimn vectors a and A are given by
a = col[a(0),a(l), , a(N)]
A = col[A(0),A(l), ,A(N)]
The (N+1) X (N+1) matrix y* is the conjugate of the DFT matrix y as given by
1
y
1 y N . . 2N
- y
(2 18)
Where
y = exp[-j27i/(N+I)]
Equation (2 17) is the IDFT and its inverse (DFT) is therefore given by
A = Ya (2 19)
It is noted that the DFT matrix Y has the property
YY =(N+I) I
16
Where I is the identity matrix
Although, in principle, the DFT can be evaluated using equation (2.15) or
equation (2.19) in practice the fast Fourier transform (FFT) algorithm is used, [32], [33].
From the foregoing it is clear that a finite-duration sequence can be expressed exactly
by samples of its ZT. Moreover, the periodic sequence obtained by sampling the ZT at
(N+1) equally spaced points on the unit circle (|Z|=1) in the complex Z-plane is identical
to the DFT. The sequence corresponding to these frequency samples is a periodically
repeated version of the original sequence, such that if (N+1) samples, of the ZT are used
no 'overlapping' or 'alaising' occurs. Thus, in general, a finite duration sequence is
represented as one period of a periodic sequence. It has also been shown that the
ZT of a pulse train is simply a power series of Z"' in which the coefficients of the
various terms are equal to the corresponding samples when a waveform is expressed in
this form, it is possible to regenerate the number sequence merely by inspection, a
number having the index n is simply the coefficient of Z"" in the ZT.
The linear filtering operation of equation (2.10) can now be rewritten using the
short hand notation of the ZT. If the sequence C(n) represents the convolution of the
two sequence a(n) and h(n), than the ZT of C(n) is the product of the ZT's of a(n) and
h(n), i.e. if,
C(/i)= f,aik)hin-k) n=0, 1, 2, ... (N+M) * = 0
then.
C(Z) = A(z) H(z) (2.20)
This can be shown considering the following expressions,
C(Z) = Y { Z aik)h(n-k)}Z " »i=0 ft = 0
Interchanging the order of summation yields
17
»=0 n^k
Letting m = n-k leads to
C{Z)=ta{k){t h(m)Z-'"}Z
hence C(z) = A(z) H(z)
However, if the DFT is used to evaluate equation (2.10), the sequences a(n) and
h(n) have to be modified. Taking the straight forward DFT of finite duration sequences
and then inverse transforming the products of their spectra is equivalent to circularly
convolving the periodic sequences created from the given sequences. To obtain the
linear convolution both a(n) and h(n) must be (N+M+l)-point sequences. This is
achieved by appending the appropriate number of zeros to both a(n) and h(n). That is
M~ zeros
lfl(0),a(l), ,a(7V),0,0,0, ,0]
N - zeros
IA(0),A(1), ,/r(M),0,0,0, ,0]
The (N+M+1) point DFT of a(n) and h(n) is then taken, multiplied and inverse
transformed to obtain the correct sequence C(n). The ZT, a familiar analytical technique
in modern control and sampled data system, thus provides an excellent tool for studies
of digital systems and signals such as pulse strains. Therefore, throughout this work use
is made of the ZT representation where ever possible.
2.3 OPTIMUM PROCESSING OF RADAR SIGNALS
The transformation, and interference effects to which a radar signal is subjected
during its path from the transmitter to the receiver will now be analyzed using Fig. 2.6. It
is assumed that the signal, although generated digitally, is analogue filtered prior to
transmission. The transmitted signal S(t) first passes through a time-invariant processor,
which accounts for the unknown round trip amplitude attenuation a, time delay x,
18
diftiUl signal
generator analogue
filter
sCnl)
Trausruiller and
antenna
transmission path
l-H I
time-invanant processor H »1
Cr*
noise \v(t) and clutter c(,t) Mt ) + c(t)
s(nT-T^e i(2 v(nT-T)-KJ)
+u\tiTj + c(nTj
rig. 2.6 Block Oiagrain of signal iiioflcl
Doppler shift v, and phase shift (j) of the signal. Such a treatment of the transmitted signal
assumes a point target (no range extent). This is a convenient assumption in analyzing
system performance. In order to avoid continual repetition several general assumptions
are made for subsequent discussions of radar signal processing techniques.
i) point targets are assumed
ii) Target acceleration is negligible, i.e.
a « >. 11^
Where a is target acceleration, X is the carrier wave length, and Ts is the signal duration.
iii) Mismatch of the envelope of the received signal and the transmitted waveform due
to high relative velocities is negligible, i.e.
2 v^c « lAVT,
Where Vr is the radial target velocity relative to the radar, c is the velocity of light,
and W is the signal band-width.
iv) All signals are narrow band.
W « f e
Where fc is the carrier frequency. Since radar returns are always immersed in
noise and interference from all kinds of objects illuminated by the antenna beam, the
receiver must be optimized in some manner. Additive interference introduced by a large
number of independent target like reflections such as composite returns from an
extended scattering region containing terrain, rain, sea waves etc., is usually called
clutter. The return signal, immersed in clutter and noise, is now analogue filtered,
digitized and processed. These approaches have been used to derive optimum processors
for radar signals,[4] ,[21], [34].
i) Signal to Noise Ratio Criterion (SNR)
ii) Likelihood Ratio Criterion
iii) Inverse Probability Criterion.
19
Any of these criteria lead to the matched filter receiver, provided the signal is
corrupted only by additive white Gaussian noise Moreover, Woodward, [21] has shown
that this type of receiver also preserves all the information in the radar return Even in
situations where matched filter processing is not optimum, for example when
interference from clutter is significant (coloured noise), matched .filters usually provide a
reasonable compromise between system performance and complexity, [35] On the other
hand, matched filter processing may be used simply because information about the target
environment to design more optimum processors is not available The problem of target
resolution and optimum detection for a specified clutter environment has been studied by
a number of authors and will not be of prime concern here, [36], , [39] Therefore,
unless otherwise specified a matched filter receiver is assumed
2.3.1 Digital Matched Filter
The characteristics of the matched filter (MF) can be designated by either a
frequency response function or a time response function, each being related to the other
by a F T Operation In the fi-equency domain the MF transfer fimction H(f), is the
complex conjugate fiinction of the spectrum of the signal that is to be processed, except
for an arbitracy scale factor and a linear phase shift
H(f) = S*(f) exp (-j27ifrd) (221)
Where S*(f) denotes the complex conjugate spectrum of the input signal S(t)
The scale factor and the linear phase shift exp(-j27ifrd) do not affect the signal to noise
ratio (SNR) and may therefore be ignored Thus,
H(f) = S*(f) (2 22)
In the time domain the corresponding relationship is obtained by taking the
IFT of Equation (2 22) This leads to the result that the impulse response of a MF is the
mirror image of the complex conjugate of the transmitted signal S(t), and the general
relationship is given by
h(t) = S*(Td-t)
20
or simply
h(t) = S*(-t) (2.23)
in discrete form, this relation is
h(n) = S*(-n) (2.24)
So far it has been assumed that the spectrum of the reflected signal is
completely known. However, even for the simplest case of a single point target, the
return signal contains two known parameters, Doppler shift v and time delay x. In
general the spectrum of the received signal is of the form
S(f) = So(f-v)exp(-j27ifc-e) (2.25)
Where So(f) denotes the spectrum of the transmitted signal. Therefore, for matched
condition, a receiver with the frequency characteristic, (neglecting a constant phase term
and amplitude factor)
H(f) = So* (f-v) exp027tfc + 9) (2.26)
is required for optimum detection. It is clear that for stationary or slowly moving
targets (v = 0) a receiver matched to the transmitted waveform is optimal. However, since
the doppler frequency depends on the range rate of the target and is not known before
hand, optimum reception of signals reflected from moving targets can not be
accomplished by only one matched filters. An optimum receiver in this case requires a
bank of matched filters with incremented frequencies, v, of the Doppler shift v in the
expected domain. This is illustrated in block diagram form in Fig. 2.7. The output of the
matched filter bank is usually applied to a square law or envelope detector and
compared with a common threshold. If any of the outputs crosses the threshold a signal
is deemed detected, and from the corresponding delay and Doppler shift the target's
range and velocity are estimated.
21
from receiver ^
1 1
; 1
^
k
k
^
to
. . . 1 MF
Centered at fc+Ci+1) r
MF Centered at fc + i Y
1 1 1
MF Ceotered at
fc 1 1 1
MF Centered at fc- i V
MF Centervd at fc - CM) y
1
1
ki
F |
• w
1 1
\-l'
\-f 1 \ 1
l-l^ 1 1 1
M^
l-l^ 1 1
ff,
p
w
' •
- •
Logjc
Circuits
and
threshold
detection
outpiJt •
(DispUy)
Fifi. 2.7 Bank of paraUel MF' s
The response of the digital MF to an input signal reflected from a target, at range
'r' and radial velocity Vr, is obtained by convolving the fiher impulse response with the
input signal
V^ir,v)= f,{SinT- T)e\p[~j2m'inT- T)\ + W{nT)}S'[-im- n)T\
= exp[-j2nvimT-T)]T.{S[in+m)T-T\exp[-j2m>nT\S\nT)+ Y.W[(n+m)T]S'{nT)
(2 27)
Where x = 2r/c and v = 2vr fjc
fc = Carrier Frequency, c = Velocity of Light
The noise W(mT) is assumed to be a Gaussian random variable with zero mean
and variance o^ Furthermore, W(mT) and W(kT), for any integer k ?t m, are
uncorrected and therefore statistically independent Equation (2 27) can be simplified
by ignoring the non-essential phase factor, exp[-j27iv(mT-1)], and by letting mT-t=T'
Thus,
ff^iT',v)= E SlinT+T')S'{nT)exp(-j2m'nT)+ Y.Wl(n + m)r\S'(nr) (2 28)
Although the input signal and the impulse response are discrete, it is noted that
the function in Equation (2 28) depends on the two continuous variablex and v Since the
radar detector at the output of the MF usually removes the phase information, the
function of interest is generally
k(7.v)|' = t.SinT + T)S'{nT)expi-j2m>nT) t-~€0
+ Cross products
YWl{n + m)T]S'{nT)
(2 29)
The first term in the equation above is the signal term resulting only from the target
reflection
\X{T,V)\" = t,S(nT + T)S' inT)c\p{-j27ivnT) (2 30)
22
The function |X(T, V)|^ is commonly referred to as the ambiguity function. The
investigation of the ambiguity function has been a field of extensive study since
its introduction by Woodward, [21]. The origin of the ambiguity function (x=0,
v=0) may be thought of as the output of the matched filter tuned in time delay
and frequency shifl to the signal reflected from the target (point source) of
interest. For zero relative Doppler shift, |X(T,O)|^ represents the squared
magnitude of the auto-correlation function (ACF) of the transmitted signal. This
is the filter response to the reflections at a different range but at the same
Doppler as the target. Similarly, |X(o,v)|^ is the response to reflections at the
same range as the target but with other Doppler shifts.
Another property which reflects the fundamental constraints of radar
signal design is the total volume under the ambiguity function. It is shown below
that this volume is independent of the shape of the transmitted waveform
CO + > i r
V= I I \X(T,V)\ dnlv (2.31) -*>->lT
Substituting the Equation (2.30) into the Equation (2.31) and integrating first
with respect to v
Sin[;r{n - m)] GO OO
f^= Z Z S{nT + r)S'(mT + T)SimT)S'(nT). «=-oo m~ -<o T7t{n- m)
Integrating with respect to t and noting that the auto-correlation of the signal is
given by
r\(n- 1/1)7] = j S ' \ ( n - m)T + r\S{T)dz no
leads to
* Sin{K(n - m)\
^ = Z Z S\nT)S{^mT\r\{^n-ni)T\. «=—«) tn—-<xi Tn(n - m)
smce.
Sin\n{,n-m)\ [O, n^m
Tn{n-ni) I f , « = w
23
received
s igna l matched t o
sub-pulse
^ors
^ers t
delay l i n e
| s ( N ) |
- / s {1*1 {
|s<M-»)|
-/6(N-l)^
'
i
+
'
| s ( 0 ) |
- / s f o ) ,
'
output
^H. 2.8 Tapped delay line MF.
r7\ii_
received
>ignal
n A/D
90°
,( ' \ »r
LO
n
s ( o )
S ( l )
s ( 2 )
1 . 1 ni
A/D
; — ^ J J
1 1
s (N-1)
S ( f J )
1
S(o)
S ( l )
S(2)
1 1 1 r 1
s(rf - i )
s _^
a u o
> « u c « u
<M
r-t
s. n
1 1 1
1 1
s. •H a.
i
r(o)
r ( l )
r (2)
1
1
1
1 1
r(N- l )
r(N\
f^g- ^.9. DFT realization of MF.
and
1 + 0 0 2 J
or
V = j:E' (2.32)
Where E denotes the signal energy.
Hence the volume under the ambiguity function depends only on the total signal
energy. This implies that any reduction of ambiguity anywhere in the (x,v)-plane will
cause it to appear elsewhere. Equation (2.32) is particularly important in clutter and
multiple target environments. The radar signal design problem can be considered,
therefore, as a process of rearranging the undesired portions of the ambiguity function
(range and Doppler ambiguity away from the origin) into region of little importance.
The second term in Equation (2.29) is due to the white noise. The
computation of the mean noise power is simplified by noting that the noise power is an
uncorrelated zero mean, random Gaussian variable. Thus the expected value of the
various cross products in Equation (2.29) are zero. Hence the mean square noise power at
the output of the MF is given by
m m
\N{T,V)\ = £ { Z Y.iVinT)S'(nT+t)W'iniT)S{mT + T)exp[-j2m>(n-m)T\} m=-co n=~<x>
CO OO
KV(r,v) = Z Y.E{W(nT)W'(mT)S'inT + T)S{mT + T)expl-j27ivin-m)T\} /fl=-00 rt=-flO
\N{T,V)\ = S ZrJnT,mT)S'inT+T)SimT + T)exp[-j2nvin-m)T\ m=-<o n=-oo
Where
r„(nT,mT)=E{W(nT)W*(mT)}=a^5[(n-m)T]
24
hence
\N(T,V)\' = a' l:|^(«r)|' =a'E = N,E (2 33) n=-«o
Thus for a given signal energy, the mean-square noise power at the output of the
optimum receiver, is a constant over the (T,v)-plane
So far it has been implied that the two variables x and v are continuous
However, from practical considerations only signals of finite duration can be
processed It is therefore assumed that the received signal duration is (N+1)T and that
the delay x and Doppler shift v are expressed as integer multiples of the sampling period
T and fundamental frequency 1/[(N+1)T] respectively with these notations Equation
(2 30) can now be written as
\X{k,lf = n-\k\
I,S(n + k)S'{n)expi-j2;rifi) n=0
k,l = 0,±l,±2, ,+N (2 34)
In subsequent chapters attention will be focused on the case for slowly moving or
stationary targets In other words, the relative Doppler spread of the targets is
assumed to be negligible (v = o) The output of the MF for zero Doppler is the ACF of
the transmitted signal and is given by
rik) = "Y.Sin + k)S'{n) (2 35) »l=0
k = 0,±l,±2, . ,±N
An alternative way of representing the auto-correlation sequence r(k;) as above is
to use the ZT technique described in section 2 2 2 Thus, the ZT of Equation (2 35) is
given by
R(z) =r(-N) + r(-N+l)Z-'+ +r(0)Z-^+ + r(N-l)Z-'''" + r(N)Z"'''
= [S*(N)+S*(N-1)Z-' + + S*(0)Z-''] [(S(0)+S(1)Z-' + + S(N)Z-^]
25
= Z-''[S *(0)+S *( I )Z + +S *(N)Z''] [S(0)+S( 1 )Z-' + +S(N)Z-''l
= Z-' S*(1/Z)S(Z) (2 36)
The coefficients of R(z) are labeled with index values running from -N to N The
kth coefficient of the complex envelope of the ACF at a time shift is given by Equation
(2 35)
From the foregoing it is clear that R(z) is an even function which has the property
of complex conjugate symmetry, that is
r(-n) = r*(n) (2 37)
The main response peak r(o) given by
riO)=t\S(nf=E (2 38) «=0
is always real and represents the energy contamed in the sequence Moreover, it can
easily be shown that
r(o)>|r(n)| n = 0,1,2, N
The MF for discrete coded waveforms could be implemented usmg a tapped delay
line as shown in fig 2 8 It is assumed that the input is at the RF carrier (or IF) and that
each delay element is an integral number of wavelengths In addition the subpulse
matched filter must also be centered at that frequency, [40] Such a processor filters the
input signal directly as a band pass signal However, for long sequences, (N+1) > 31)
the bandwidth of the delay line presents a practical problem in that its N cascaded
stages must have an overall bandwidth > 1/T, the reciprocal of the subpulse duration
Therefore, it is often preferable to process signals at base band (zero IF or homodyne
receiver) particularly when digital implementation is required
In a typical digital processor the delay lines are replaced by digital memories.
Fig 2 10 In this configuration the RF or IF signals are heterodyned to zero carrier
frequency with a single-side band or quadrature mixer whose two vedio output
represent the in-phase I, and quadrature, Q, components of the signal (sec 2 2 1) It is
26
XT » m 9 XT » pi
even possible to add minor Doppler shifts (if known) to the Local Oscillator (LO) to
prevent degradation of the matched filter output The base band signals are then low pass
filtered, sampled and converted to digital form at high speed The digital matched filtering
operation can be carried out using a digital tapped delay line (transversal filters) as
shown in Fig 2 10, or by the high speed pipe line FFT method, [41], [53], [54], [55],
shown in Fig (2 9) However, in most radar application the FFT realization of the filter is
more desirable This is particularly the case for signals with a large time-bandwidth
product, since FFT methods tend to be more efficient at performing convolution
operations
2.3.2 Pulse Compression Radar
It has been shown that, regardless of the signal shape, the MF produces one global
maximum output value which is equal to the signal energy Because of this concentration
of the entire signal energy its detection against a white noise background is
enhanced in addition, to achieve high accuracy and resolution of radar measurements, it is
required that the maximum value in the MF output be as narrow as possible Since the
sharpness of the MF output signal (auto-correlation Sanction) is inversely
proportional to the rms signal bandwidth compression of the received signal into a
narrow spike can be accomplished provided the signal has a large bandwidth Thus the
essence of pulse compression radar systems is to provide this large bandwidth without
degrading radar performance in other respects such as range resolution
An obvious way to achieve a large bandwidth is simply to reduce the duration
of the transmitted pulse However, since target detectability and measurement precision
depends on the signal energy The transmitted power must be increased proportionally,
to keep the energy constant Unfortunately, the peak power limitation of transmitters sets
a lower limit on the pulse duration Therefore, the need for a large bandwidth must be
met by inodulaliitg tiie pulse, ratlicr than teduciiig the pulse duration
In principle any of the three basic types of modulation could be used to increase
the signal bandwidth, namely amplitude (AM), phase (PM), and frequency (FM)
modulation (Here, FM is considered as a general type of modulation where the phase of
the signal is varied) AM, however, is generally not desirable for use with radar
27
waveforms due to its inherent disadvantages First, it is an inefficient way of
increasing the signal bandwidth, is that the function actually applied to the
modulators must be efficiently under constant amplitude conditions Thirdly, it is
expensive and difficult to achieve good amplitude linearity throughout the radar system
over the entire dynamic range of interest Therefore, AM is of interest as a means to
improve system performance, rather than as a primary method of achieving large signal
bandwidth The case of quantized FM will not be considered here as, due to complexity in
frequency synthesizing it is less practical, except in a few cases, than PM
As implied by the term pulse compression the objective of the receive filter is to
compress the received long pulse having a time duration of T, seconds and a bandwidth
of W hertz into a short pulse of duration 1/w, to allow recognition of closely spaced
targets. The ratio of the duration of the long pulse to that of the short pulse is called the
compression ratio Thus the compression ratio is given by
mc = Ty(l/w) = TsW (2 39)
which is equal to the time-bandwidth product of the waveform In a digital system the
signal duration Ts is equal to (N+1)T, where T is the sampling interval and (N+1) is the
number of samples Furthermore, if the sampling process is carried out at the Nyquist
rate T = 1/w, then
mc = (N+l)TW-N+l (2 40)
that is, the compression ratio or time-bandwidth product is equal to the total number of
samples
The main conclusion from the foregoing is that in all cases where the transmitted
signal spectrum is substantially widened by modulation, decompression of signals can be
accomplished during reception
2.3.3 Range Resolution in a Matched Filter Radar
The estimation of target resolution performance is probably the most difficult
problem to solve in modern high performance radar systems In some cases the
interfering objects may themselves be targets of interest, whereas in others they may be
28
undesirable scatterers introducing a type of noise, known as clutter, into the system
As mentioned in section 2 3 1 the optimum receiver for maximum resolution is not
necessarily a MF In practice, however, the typical target situation is too complex and not
enough prior information is available to implement anything but a MF receiver or an
approximation For good target resolution, it has thus been necessary to retain a MF
processor but to optimize the signal waveform so as to reduce the mutual interference
(self-clutter) between targets
The ways in which the transmitted waveform limits the radar performance in white
noise are wdl known, [21] Three parameters are of prime importance These are the
bandwidth W, the time duration Ts, and the total signal energy E Specifically
i Range resolution for a MF receiver and stationary targets is determined by the
spectrum envelope of the For a given spectral shape, range resolution is
proportional to lAV Therefore, good range resolution is achieved with a
spectrum for which the total occupied fi"equency band is large
ii Making use of the time-fi"equency duality, velocity resolution (radial) for
targets having the same range is determined by the time structure or envelope of
the signal For a given envelope, velocity resolution is proportional to 1/Ts
Hence low velocity ambiguity requires a waveform that occupies, with
significant energy, a large total time interval
iii Target detectability is determined by the ratio of received signal energy to
received noise power (SNR) For given system parameter, signal detectability,
and thus range, can only be improved by increasing the transmitted energy
Therefore, it is desirable to transmit a waveform which has both a rectangular
envelope as well as a rectangular spectrum
In order to appreciate the resolution problem consider two stationary targets
slightly separated m range Neglecting an amplitude attenuation factor the combined
received siunal is of the form " C
S(nT) = So(nT - Xo) + So (nT - to - x) (241)
29
where
So(nT) = transmitted waveform
To = position of first target
X = target separation
To be able to distinguish the two target returns it is necessary to select a
suitable waveform So(nT) A measure of distinguishability, given by the sum of the
square difference of the two signals, can be expressed as
CD
f ' = I,S^inT-Tj + S^inT-T„-T)-2ReS\nT-T„-T)
£'= t {\S{nT-T,)\'+\S{nT-T,-T)\'-2Rt[SinT-T,)S'(nT-T,-T)]}
The first two terms represent the signal energy E and are therefore constants
The last term is recognised as the ACF r(x) For the two signal returns to be as different
as possible it is required to maximize the equation above, that is
max E = {E - Re[r(x)]) (2 42)
Hence, in a MP radar the optimum waveform to use would be one whose
receiver response, or ACF, has an envelope consisting of a single spike at x = 0 of a
width smaller than the spreading of the targets in range (delay) However, in practice
such waveform can not usually be realized Actual waveforms have ACF's whose
envelopes show one or more of the properties indicated in Fig 2 11 Basically there are
three types of MF responses, a single lobe (a), a narrow main lobe accompanied by
relatively large side-lobes, (b), and a single lobe surrounded by a noise-like, low-level
response spread out in time (c) Each one of these response types presents its own
resolution problems, suppose the MF output consists of a single lobe (a) If the separation
of the targets is larger than the width of the lobe there will be no problem in resolving
them However, for closely spaced targets the responses overlap and the envelope of the
combined output will depend on the phase relationship of the echoes. Fig 2 12 Thus, in
the region of overlap, target resoJutJOJo s difficult to achjeve Therefore, to distinguish
two or more targets their returns must be separated by at least the half power (3dB)
30
(a)
(b)
A
| r < t ) |
A
(c)
Fig. 2.11 Forms of range aobiguitles.
MP outpuc
target 2
' o * ^
MF output
T >1/W
O O
MF output carriers in phase
carriers out of phase
T » 1/W
Fig. 2.12 Overlap of two target returns.
response width A diflferent type of resolution difficulty is caused by the MF response
shown in Fig 2 11(b) Although the main lobe may be sufficiently small to meet the
required target resolution, target returns separated at multiples of Ta are completely
masked Problems of yet a different nature are introduced by the response of type (c)
Again the main lobe may be narrow enough for the desired close target resolution
However, for targets with widely varying cross-sections it still may not be possible to
distinguish them. The pedestal like extension of the response due to a strong target
may have an amplitude strong' enough to obscure the main response peak of weaker
targets. This effect is aggravated particularly in a multiple-targets environment where
the combined side lobes from many returns may build up to a level that even relatively
strong targets can no longer be recognised
As shown by Woodward, [21], the MF receiver utilizes the full information
available from the return signal The width of the main response lobe can be regarded as
a measure of uncertainty about the exact target range, while the spread of the response
introduces ambiguity of the target location Both effects, although conceptually different,
are lumped together in a figure of merit known as the time resolution constant, [21]
^T = — - — (2 43) k(0)|
The resolution problem has been discussed here in a purely qualitative manner
Analytical treatments of the resolution in parameters can be found in many excellent radar
books, [22], [23], [26], [42] Nevertheless, the preceeding discussion allows the
general formulation of the requirements which have to be met for target resolution First,
the output signal to white noise ratio must be large enough for reliable detection
Secondly, the target of interest must be separated sufficiently from any other target of
comparable or larger cross-section to prevent overlap of the main response lobes
'I'hiidly, the combined intcifeionce liom othei taigcts must not be so strong as to mask
the target return of interest
31
Interference from other targets due to response sidelobes acts like clutter caused
by undesirable scatterers. However, the term clutter implies that the interference causing
reflectors are so dense that they can not be resolved. The type of clutter due to side lobes
is often called self-clutter to distinguish it from the effect of undesired objects.
In summary, the resolution performance of a radar thus depe'nds not only on
the width ofthe main response lobe but also on the low level response surrounding the
main peak. In the published literature resolution is often referred to as the 3dB points
of the main response lobe. In the present context, however, target resolution means
the ability to recognize a target in the presence of others.
32
CHAPTER-3
SYNTHESIS OF PULSE TRAINS FROM SPECIFIED AUTOCORRELATION
FUNCTIONS
33
3.1 INTRODUCTION The problem of synthesizing signals which realize the desired auto-correlation
function (ACF) can be divided into a number of inter-related problems, each of
which has an independent practical significance The separate problems could be
formulated as follows, [42]
1 Determine the class of functions which are realizable ACFs for arbitrary signals
2 Determine the subclass of ACF's for various signal structures such as discrete
coded waveforms
3 Synthesis of signals whose ACF is a close approximation to the desired ACF not
belonging to the class of realizable functions
4 Synthesis of signals (pulse trains) which satisfy a given set of requirements,
e g range resolution, energy utilization, etc
So far, a comprehensive analytical treatment of these problems and their
solution has not been formulated For example, a simple criterion for determining the
realizability of an ACF has not yet been found
To appreciate the nature of the problem arising here consider the ACF
rit)= TSinT+T)S'{nT) (3 1) /I=-0O
or its equivalent expression
r(r) = f\Sif)\" txjp(jl7tfT)df (3 2)
Where |S(f)| is the power spectrum of the signal S(nT) and is assumed to be
band limited In other words S(f) is zero outside some range (-w/2, w/2) strictly
speaking the requirement for a finite band width W is incompatible with a finite duration
signal However, an approximation to the finite spectrum condition can be reached if
a major portion of the signal energy is concentrated within a specified frequency band
34
From Eq (3.2) it can be seen that the ACF and the power spectrum form a
Fourier transform pair Hence it follows that for r'(x) to be a realizable ACF its spectrum
R(f), must be real and non-negative. Even if the given ACF is realizable the synthesis
problem can not be solved uniquely Since the phase information is lost in the power
spectrum, it is not possible to determine S(f) itself which is necessary to find S(nT)
Therefore, all signals whose spectra differ only in phase will have the same ACF Thus
the synthesis problem may be divided into the following two steps
I. The power spectrum IS(f)| is determined from the given ACF (It is assumed that
the ACF is realisable, i e, R(0 = FT{r(T)}>0
2 From the determined power spectrum one signal having such a spectrum is derived
by assigning an arbitrary phase fiinction 9(f), i e,
S(nT) = lFT{|S(f)|exp(ie(f)}
For digital application, however, only finite length sequence can be processed
The next section will, therefore, be devoted to the problem of factorising the power
spectrum using ZT techniques
3.2 SYNTHESIS OF PULSE TRAINS IF THE ACF IS KNOWN IN MAGNITUDE AND PHASE
If the ACF is given in phase and magnitude at discrete points its ZT can be
written as Eq (2 36)
R(Z) = Z- S(Z)S*(1/Z)
As mentioned previously the ZT provides a convenient method to represent a
signal in the form of its zero pattern which is obtained by factorizing its polynomial in
Z In factorized form the equivalent representation of a polynomial S(z) of order N in
power of Z"' can be written as
SiZ) = SiO)h(l-^) (3 3)
35
Im[z]
vinit circle
Re[z]
«»
7^ O)
F i e . 3 . 1 (*) Factorized ACF of 11 element Bairkei Code (b) RoBulttng sequence when choosing the zero p&tletnl,2,3, ,10
Where Z, are the zeros of S(Z), i e S(Z,) = 0, i = 1, 2, , N) Similarly, S*(l/Z)
can be represented as
N
u 1=1
s'(i) = s'{o)Uii-z.z;) (3 4)
S'{^) = S'iN)U(Z-jr)
R(Z) = S{0).S' {N)Yl (.1 - ^)iZ - jr)
Where the unessential delay factor Z" has been neglected Since R(Z) essentially
represents a power spectrum the above equation can be regarded as the factorized power
spectrum, [43]
The equivalent expressions above allow the study of pulse trains using their
zero patterns in the complex Z-plane The conditions S(0) ^ 0, and S(N) ^ 0, are clearly
equivalent to
S(0) ^ 0 and S*(0) ^ 0
It is easy to verify that if S°(Z) devotes the polynomial
S"(Z) = Z'^S*(1/Z)
then
(S''(Z))°=S(Z) (3 5)
and
(S(Z) P(Z))" = S"(Z) P"(Z) (3 6)
In addition it follows from Eq (3 3) and Eq (3 4) that for
S(Z) = |S*(1/Z)| (3 7)
Moreover, if a polynomial S(Z) of degree N has Pi zeros inside the unit circle,
|Z| = 1 (counting multiples), P2 on the unit circle and P3 zeros outside, where
Pi+P2+Pi=N, it is referred to as of the type (Pi, P2, P3) Since it has been assumed
36
that S(0)5tO, it is clear that Zj is a zero of S(Z) if 1/Zj' is a zero of S*(l/Z). The zeros Z
and 1/Zj have the same angle in the Z-plane but reciprocal magnitude as indicated in
Fig 3 1 It is clear that S(Z) is of the type (Pi, P2, P3) if S'(l/z) is of the type (, P3, P2, Pi)
The class of polynomials, P, for which P(Z) and P*(l/Z) have the same set of zeros
are known as self-inversive polynomials, [46] It is apparent that a polynomial P(Z) is
self-inversive if its zeros are symmetric with respect to inverse on the unit circle from
the foregoing it should be clear that
1. A self-inversive polynomial of degree N is of type (P, N-2P, P) for P > 0
2. Since the polynomial R(z) consists of the product of the two factors S(z) and
S*(l/Z) it is of type (P1+P3, 2P2, P1+P3), where 2(Pi+P2+P3) = 2N Consequently,
R(Z) is self inversive and its zeros must occur in reciprocal conjugate pairs
Thus for a finite pulse train to be an ACF it has to satisfy condition (ii) The
design technique for pulse trains from a given realizable ACF can now be summarized as
follows
1 Factorization of the ZT polynomial which represents the ACF
2 Selection of a suitable zero pattern to obtain the signal after multiplication
The design procedure is probably best illustrated by an example Consider the 11-
element Barker Code where ACF has the ZT representation
R(Z) = -i-z-^-z-^-z-*-z-*+i iz-'^-z-'^-z-'^-z-'^-z-'^-z-^"
This polynomial can now be factorized on a digital computer using a standard root-
finding algorithm The resulting zeros arc given in Fig 3 1(a) All the zeros occur in
reciprocal conjugate pairs In addition, as a consequence of the coefficients of R(z) being
real, all complex zeros must occur in conjugate pairs The selection of zeros for S(z) and
multiplying them out completes the design procedure The resulting sequence choosing
the zeros labeled as 1,2,3, , 10 is shown in fig 3 1(b)
37
3.3 SYNTHESIS OF PULSE TRAINS IF ONLY THE MAGNITUDE OF THE ACF IS KNOWN.
In the previous section the magnitude and phase of the ACF at discrete points
was required to find a solution to the synthesis problem However, fi-om practical
considerations only the magnitude of the function is usually known, since the phase does
not effect the accuracy and resolution of the range measurements In this section the
design procedure is extended to the case where only the magnitude of a realizable ACF is
given at discrete points The basic underlying idea of the method presented here is
due to Vakman47 and is afso implied by VocJcker, [45J in a different context
Before proceeding any fiirther it is necessary to recall the convolution theorem
derived from basic Fourier transform theory
r(t)* r*(-t) ^ |R(f)r (3 8)
|r(t)|^-^R(f)*R*(-f) (3 9)
The above relation show the duality between the ACF of the spectrum R(f)
and the ACF of the time signal r(t) As r(f) is assumed to be band-limited it can be
represented by its Nyquist samples However, due to the convolution process in the
frequency domain the squared envelope, |r(t)|^, will have twice the band-width of r(t),
[29] In other words, if |r(t)p is sampled at Nyquist rate, r(t) is sampled at twice that rate
Hence it is assumed that the squared envelope of the ACF is known at integer
multiples of T/2 = T'
Since |r(t)| is of finite duration it is completely defined by frequency domain
samples taken at intervals 1/T' Such a signal has a finite Fourier representation of N
terms
init) = |r(Or = j:c(k)expU2nkir) (3 10) k U
where Ts is the duration of the signal and N = WTs
The samples of the envelope are thus given by
38
m(^)=tc(k)txpij27tk-^) (3.11)
n = 0, 1,2, ....,(N-1)
By substituting the symbol Z for exp(j27it/T.) the finite Fourier series can be a
rewritten as a polynomial in powers of Z
m(Z) = C(0) + C(1)Z +C(N-1) Z""-'
m{Z)= Zc(*)Z* (3.12)
This transformation can be regarded as the dual of the ZT discussed in
Chapter 2.
Consider now the polynomial of order L representing the ACF, r(t).
r(Z) = R(0) + R(l)Z + +R(L)Z^ (3.13)
clearly, for r(Z) to be realizable all R(n) must be real and non-negative (R(n) > 0), since
the coefficients of the polynomial are the power spectral samples. The squared modulus of
r(t), |r(t)|' can thus be represented as a polynomial multiplication.
m(Z) = [R(0) + R(1)Z + + R(L) Z'-J-LRU) + R*(L-1)Z +...+ R*(0)Z ]
m(Z) = Z^[R(0)+R(1)Z +...+ R(L)Z^].[ R*(0) + R*(1)Z-' +...+ R*(L)Z- ]
m(Z) = Z^ r(Z) r*(l/Z)
Where the coefficients of m(Z) are given by
L-\k\
C(A)= £/?(/i)/?(/i + A) «=0
k = 0,±l,....,±L
Since r(z) is a polynomial with real coefficients
m(Z) = Z^ r(Z) r(l/Z)
Thus the operation of convolution in the frequency domain reduces to a
multiplication of two polynomials. This clearly reflects the duality of time and
frequency as pointed out earlier.
39
Since m(z) has the form of an ACF it is possible to proceed in a similar manner to
that described in section 3 I in order to find the power spectral components R(n)
given m(z). The properties of m(z) are revealed by studying the zeros in the complex
Z-plane. The coefficients of m(z) specify the ACF of the spectrum of r(z). Its 2^ zeros
must therefore occur in reciprocal conjugate pairs. In addition, since all coefficients are
real (and in particular non-negative) they all occur in complex conjugate pairs. Thus if
Zj is a zero of m(Z), then Zj, 1/Zj and 1/Zj* also must be zeros of m(z). This relationship is
illustrated in Fig. 3.1(a). Consequently, if m(z) is to represent a realizable power
spectrum ACF the following conditions must be satisfied.
i) m(Z) is finite and its zeros occur in complex conjugate reciprocal pairs,
ii) The coefficients, R(n) of r(Z) must be real and non-negative.
If these conditions are met then at least one and in general a whole set of ACF's
having the same magnitude can be found. The steps in the design procedure can be
summarized and best illustrated by using the 7-element Barker code as an example.
1. Given the samples of r(t), where r(t) is assumed to be band limited and sampled
at twice the Nyquist rate. Fig. 3.2(a), one computes the DFT of the sequence
|r(Kr)| i.e.
DFT[|r(0)p, |r(r)p .... |r(13T')|^..., |r(T')ft
This gives, except for a scale factor, the (2L+1) Fourier coefficients of the
periodically repeated square envelope of the ACF, Fig. 3.2(b).
2. Factorize the polynomial whose coefficients are these Fourier components. The
2' roots should occur in reciprocal complex conjugate pairs. Fig. 3.2(c)
3. Select L roots fi'om each reciprocal conjugate pair and its conjugate. Now,
multiply out to obtain a set of (L+1) Fourier coefficients which are, neglecting
a scale factor, the DFT of the samples of r(t). Verify that r(t) is indeed an ACF.
This is done simply by making sure that the coefficients obtained are all real and
40
3.2(tt)
3.2(b)
3.2(c)
1
iT..n.ti -I3r»
|x(nT*)|
1
JLJI^ 13T'
nrr{ | r (nT*) | ' ) 1
iL —t -iy2T
K 1/2T«
x-plan«
I Dnp{r(nT))
(d) ' ' f 7 f F f ' 9 9 9 9 '
-V2T O 1/14X 1/2T
(e)
r(nT)
i./fe:
LLJ «—I o '—•- ULA TV r
7T
rig. 3.2 (a) Sampl«s of th« ACP magnitude
U>) Spectrin sanples of ssLtiared •nv«lop« of (a)
(c) Zero pattern of (b)
(d) Spectrum saaples of the ACT
(e) ACT in Magnitude and phase .
non-negative. If the test fails, select a new zero pattern and repeat the procedure
from step 3, until a realizable ACF is obtained, Fig. 3.2(d). From the L roots
only L/2 can be chosen independently, since the zero must be selected in
complex conjugate pairs. Hence, there are in general 2V2 possible zero
patterns. However, not all zero combinations will result in realizable ACF's.
The zeros chosen in this case are labelled 1,2,...., 13, Fig. 3.2(c).
The final stage of the synthesis procedure is to take the IDF of the Fourier
Transform coefficients to'obtain the sampled values of r(t) Fig. 3.2(e). Once the
ACF has been found in magnitude and phase, it is then straight forward to
synthesize a pulse train which realizes this ACF by following the procedure
outlined in section 3.2.
3.4 DISCRETE PHASE APPROXIMATION TO LINEAR FM SIGNALS
The Linear FM (LFM) or (Chirp) waveform is probably the principal type of
signal transmitted by a radar or sonar system. The main advantage in using such
waveform lies in their case of generation and insensivity to small doppler shifts. The
description and properties of analogue chirp techniques are well documented in the
literature and will not be repeated here, [22], [25], [41].
In general the complex envelope of a LFM signal can be expressed as
S(t) = exp Ol tV2) (3.15)
where
H = 27iw/Ts
W = fz - fi = Frequency Change During Sweep
Ts = time duration of the sweep.
41
As implied by the term LFM, the instantaneous frequency is swept linearly from fi
at t=0, to a maximum value of f2 at t=T. The complex envelope can be written in terms of
the pulse compression ratio (time bandwidth product) denoted as nic
S(t) = exp[J7t(wt) /mc]
If the waveform is sampled at uniform time interval of T seconds
S(nT) = exp[0'7i(wnT) /mc]
and with
mc = NTW
S(nT) = exp07tWTn /N) (3 16)
n - 0 , 1,2, , (N-1)
The total phase change over the signal duration is
(1) = 7CWT(N-1)VN
Furthermore, if T is set equal to the Nyquist rate, then
T=1AV
and
S(nT) = exp OTtn'/N) (3 17)
Since each segment is coded into one of N possible phases these sequences are
some time referred to as polyphase codes In particular the sequences whose phase
follows a quadratic progression will be called Quadratic Phase (QP) codes An
interesting property of QP codes is their periodic ACF which is zero for all non-zero time
lags, [50], provided the sequence is coded as in eqn (3 17) if N is even and is modified to
S(nT) = exp (J7cn(N+1 )/N) (3 18)
for N odd
42
Higher order poly-phase codes with zero circular ACF (k ^ 0) have been described
by Frank Zadoirand Heimller, [16], [17] & [18], The length N of such codes, however, is
restricted to perfect squares. While the QP sequences and Frank codes have ideal
cyclic auto-correlation, they do not have of course perfect a periodic auto-correlation.
Another property of practical importance is the simple generation of QP pulse
trains if the sequence length is chosen properly. This can be demonstrated by expanding
the expression n /N as
n /N = q + qi + a„/N
where q = 0, 1, 2,
qi = 0 or 1
a„ = remainder
f — 1; q.is.odd
+ 1; q.is.even
and
exp (J7i/2)=j
The QP code can be also written as
S(nT) = (±l)(l/j)exp07ta„/N) (3.19)
The number of dilTcrent samples to be generated is thus a function of the
number of distinct remainders of n^/N. Roy and Lowenschurs, [51] have shown that for a
proper choice of N the number of different samples can be kept very small indeed. For
example for N = 16 only three different values must be generated, exp(j7i/16), exp(J7i/4)
and 1. Incidentally, this property has also been exploited in the Blusteen algorithm which
computes the DFT using a Chirp Filter, [52].
43
3.4.1 Properties of the Compressed Pulse Train
The exact expression for the ACF can be obtained by substituting eqn (3 16) into
Eqn (3 1)
rikT) = ""ZexpOVnv^.I/i' - ( / i + ^ ) ' l 11=0
J N-l-K
k = 0,1,2" ,(N-1)
and can be written in closed form By rewritting the sum term as Y,f", Where r =
The summation in the last expression is of the form of a geometric progression
N-l-k
Z/ n=0
exp(-j27iWTK/N) It is recognised that the series containing a total of (N-K) terms has
r"-' -1 a sum of, 5 = r-\
Therefore, •„ . 2 7 . exp|-72;nt>7A(N -k)\-\
txp{-jlmvT N)-\ r{kT) = c\p{-jmvk^i) ^ . - , _ . ^*
Since only the magnitude of the ACF is of interest
r(kT)\ \Sin\mi>Tk^^i
\Sin{mvT "si
If T is equal to the Nquist sampling rate, i e WT = 1, Eqn (4 16) becomes
Because r(KT) exhibits complex conjugate symmetry with respect to k = 0, it is
sufficient to consider only positive time lags The nature of the function (3 21)in the
vicinity of k = 0 has the form of a sine function with a peak value of N Because of the
periodicity of the expression this characteristic will be repeated at intervals 1/N For even
length sequences the function is symmetrical with respect to N/2 The effect of the term
(1 -K/N) can be explained, considering that
44
|Sin[7iK(l-K/N)]| = |Sin(7cK'/N)|
Thus it modulates the frequency of the ripples in a 'chirp like' fashion. Whenever
a sequence is used as a modulating function it is always of interest to consider its
spectrum. Since the power spectrum contain the relevant information, it is sufficient to
consider the amplitude spectrum only. For convenience the spectrum is assumed to
be zero out side some range (0,W). This is no restriction, since any band limited signal
can be brought into this form by a suitable frequency translation.
The magnitude of the ACF is shown in fig. 3.3 for a QP sequence of length N =
128 when sampled at the Nquist rate, WT = 1. The characteristics of the QP pulse trains
when sampled at a lower or faster rate than the Nyquist rate (under or over
sampling) are depicted in Fig. 3.3 & 3.4. These graphs reveal some interesting
properties. First, if sampled at the Nyquist rate (WT = 1), the ACF consists of a sharp
narrow spike with low residue side lobes. Secondly if WT < 1, over sampling occurs and
side lobes near the main peak appear. This is not surprising since increased sampling
rate implies a closer approximation to the analogue FM signal whose maximum side-
lobes are immediately adjacent to the main lobe, as shown in Fig. 3.4. In other words, the
sampling points do not miss these large side-lobes as is the case for WT = 1. Thirdly, for
under-sampling WT > 1, the aliased versions of the ACF will produce significant range
ambiguity at time shifts K = NAVT from the compressed pulse i.e. for WT = 2, K = N/2
as illustrated in Fig. 3.7. The cause of the spurious response peaks can be avoided
provided the waveform is sampled at the Nyquist rate. The resulting sequences have low
side-lobes and low side-lobe energy and thus are suitable in a multiple-target
environment. However, there may be specific cases where WT is made slightly greater
than one, to attain somewhat better range resolution. By comparison of Fig. 3.6 with 3 3,
it can be seen that a small increase in WT tends to reduce the side-lobes close to the main
45
lobe at the expense of an increase towards the end of the response The important
properties of QP pulse trains where WT = 1 may now be summarized as follows
i) If sampled at the Nyquist rate QP codes have virtually all the properties of LFM
signals Their ACF consists of a single spike with low level side-lobes
ii) The codes have zero periodic ACF's for K ;t 0
iii) For WT < 1, large close-in side-lobe appears, whereas for WT > 1 spurious
response peaks occur further away from the main-lobe
iv) The major side-lobes occur in two bands approximately centered at time shifts
K = V(N/2)and K = [N-V(N/2)] respectively The width of the bands is about
2[>/N - V(N/2)] -Moreover, for sequences of even length N, the side-lobe
structure is symmetrical with respect to K = N/2
v) The maximum side-lobes increase approximately as 0 SVN and the energy ratio
E ,<5%forN>40
vi) Relative simple generation of such pulse trains with a suitable choice of N
Hence QP sequences have good range resolution properties which make them
suitable for a dense-target environment For example, the peak range side-lobe for N =
128 is -27 5 dB down on the main response and the r m s side-lobes are about -36 6 dB
down
In subsequent chapters an essentially different method of synthesizing discrete
coded signals with desired auto-correlation properties is described The method is
based on numerical optimization techniques Such an approach has a number of
advantages First, no restriction on the class of admissible phase functions is
necessary Secondly, no information of the signal's phase structure is usually required
Thirdly these methods are flexible in a sense that it is possible to control particular side-
lobes or side-lobe regions
46
l.o—I
0.5
, l'<^)|
X -13dB
64T
(a)
wp _ n «
- ""T 128T
I.O
|s(f)
0.5
Fig.-> 3.3—(a) ACF, (b) an^Iltude spectrum of
12Q-element fiP codo for WT • 0.5
F i g . S. l ACT o f a LFM s i g n a l .
l.O _||xlT)|
I U)
mr - 1.03
0.5 —
,close-in sidelobes are reduced due to slight undex-sanpllng
-22.7d8
1 64T T 128T
tb)
1/2T l A
Tig. 3.6 Effects of sli9ht under-sampling on the
(a) ACF, and (b) the amplitude spectrum.
l.O Ir(T)l (a)
NT - 2
0.5
k^ooft.
/
•putlous response sptX* dua to aliasing
^orW^ t'^V.-.ftft^.
64T 128T
l.O
|S(f)|
0.5
(b)
wr - 2
1/2T I/T
Fig. -3.7 Alla-;lng effects on the ACF (a), and Anplltucle spectrum (b), doe to
CHAPTER - 4
DESIGN OF BINARY CODED PULSE TRAINS
47
4.1 INTRODUCTION
A relatively large group of signals which have received special attention are
certain binary sequences (Group I, Chapter 2) Such signals have been extensively
considered for improving ambiguity and resolution in radar and solving special problems
in the field of communications, [5], [7], [8], [14] and control system, [75]
The attractive feature of binary coding is that a number of simple, efficient and
flexible decoders can be built (a pair of shift registers can be used as a tapped delay
line pulse compressor) The aperiodic ACF of a N-element binary sequence C(n) can be
written as
r(A)= ^c{n).c{n + k) (4 1) »i=0
Where C(n) = ±l, n = 0,1,2, ,N-1
A class of binary sequences whose ACF's satisfy the conditions
r(k) =
+ N; if.k^O
0; if .{N - k)J\.cxven (4 2)
± 1; if.(N - k).is.odd
are called Barker sequences or perfect words, [8] Barker Sequences exist for length N =
[2, 3, 4, 5, 7, 11, 13] Turyn, [9] has shown that there are no other binary sequences with
this property for 13 < N < 6084 and that it is unlikely any will exceed 6084 The
limitations to a maximum length of 13 is a serious one in radar detection Consequently
considerable effort has been devoted to the problem of finding longer binary sequences
which, if not optimum, are at least satisfactory for a given application, [6], [10],
[11]. [14] for a binary sequence of length N There may be 2N possible combinations
and for the binary sequence of length N will best possible ACF, the computer has to
search 2N these possible combinations For large values of N the searching problem
becomes enormous J Linder, [56] has given sequences of best possible auto-correlation
function upto the length 40
It is well known that by choosing a large number of elements C(n) randomly,
sequences where r m s side lobe levels are of the order VN can be found However, it
48
can be expected that in the statistical synthesis a large number of side lobes will exceed
VN. It will be shown that by proper choice of the sequence both average and peak side
lobe can be held at a lower value and thus yield a better range resolution and clutter
rejection However, nothing is known about how small the maximum peak side lobe
might be in the best cases. At present there is apparently no solution, other than a
exhaustive search, to this general problem and good binary sequences which approach
the Barker codes have been found only by trial and error.
The major difficulty in synthesizing binary sequences i) the discrete nature of the
amplitude and phase. Consequently, the major objective in subsequent sections will be to
describe the various ways of obtaining binary coded waveforms using Numerical
methods, to study their properties, and to consider where they are suitable.
4.2 THE OPTIMIZATION PROBLEM
The discrete nature of pulse trains complicates their synthesis considerably.
No ordinary methods can be used in an attempt to design such sequences unless a
discrete phase approximation to analogue signals is made. For example, the methods
based on the stationary phase principle results in relatively large side lobes (> -30 dB)
and moreover, limits the class of admissible waveforms. Instead of synthesizing pulse
trains from a given power spectrum one can try to find the actual signal itself which, if not
ideal, has at least a satisfactory approximation to the desired ACF. In general,
approximation is essential, since the specified ACF is rarely realizable for a given set
of constraints on the signal waveforms. After an initial solution has been obtained. It
is compared with the required MF response. The result of the comparison is the
appioximation error and the objective is to reduce the error by modifying the initial
solution.
Before presenting the optimization techniques used on the class of signal design
problems considered here it is necessary to state the design problem in a suitable form.
49
4.2.1 Forniiilntion of (he Synthesis Problem
The most important steps in the design procedure may be outlined with
reference to Fig 4 1 as follows
First, at the outset the designer should have a clear under-standing of the
functions to be performed by the signal This step includes an evaluation of the specific
radar tasks such as multiple target resolution capability, transmitter power
limitations, etc
Next, the principal waveform type has to be selected Here the designer has to
decide whether to use signals of Group I, II, or III (Chapter 2) The signal under
consideration may be characterized by means of a set of design variables in amplitude,
(a„), phase (|)„, and frequency f„ It is often convenient to replace ordered sequences by
vectors to permit the rotation of linear vector-spaces Thus
X - (|a.,|,|ai(, ,|aN-i(,(j).„(j)i, ,(|)N-i,Cri, SN- 0 ' (4 3)
where x is the design vector
The next step is the mathematical formulations of the system This is expressed
as a set of equations
hj(x) = 0 , f = i , 2 , ,1 (4 4)
In most practical cases there are restrictions on the permissible values of the design
variables which may be written in the form
g.(x)>0, i = l , 2 , ,m (4 5)
These constraints exclude undesirable solutions for a particular application All
solutions which simultaneously satisfy Eq (4 2) and (4 3) correspond to acceptable or
feasible designs At this point a criterion or objective function must be adopted to
determine when the 'best' solution is arrived at for a given degree of complexity There
are several alternative ways to define a measure of best performance For example one may
choose to minimize the sum of the squares of the errors, the absolute error, the sum of
the absolute errors and so forth The choice of any of these criteria is usually dependent
50
-
••
Pormultttion of syst«a requlrenents ' '' •••'•^ - <
• • :•••••• 1 -
Selection of wavefon type
J - Definition of design variables! x - QaJ*• • • #J»i,_il •
}O'-**'+M-1''O'***''M-1*
•
Mathematical formulation of' the behaviour of the systeat
h (x) - O, J - 1.2 A .
• *
Constraints a. 9. (£) O
i •• 1(2,...,m
Selection of the performance
indext P(x)
Choice of an initial
design I 3C
<M
o
Control of the constraints
I Evaluation of F(x)
I F(g) »« min
Final evaluation I pract ica l considerations (conqplexity)
Optiatal so lut ion
b n_z
Select ion of new design
_(Jt )
{^
Fig. 4 I Fl'jw r'lTcir.T'i of thf doj»inr proccju-e.
on many factors, the principal ones being the tasks the system has to perform In fact if
the optimization technique is adequate, the performance index will completely determine
the final system
At this stage a method to search for the optimum solution remains to be chosen
Several techniques exist for minimizing non-linear functions, [64], through [73] All
these methods have been investigated in details and their convergence efficiency has
been studied, [72]
The function of interest is the sampled squared magnitude of the MF response
N-l-k
rikf |2
IrC" I^««««+* (4 6)
The aim is to find the design vector X = (|a|(j)) which minimizes the objective
function or performance index (also known as error criterion or cost function)
F(x)= t f{Hkf-\r(kf} (4 7)
Where f(A)is the desired ACF a n d / i s an arbitrary but suitable function of
interest For most practical applications the signal will be subjected to the constraints
|an|=l, n = 0,1,2, ,(N-1) (4 8)
Eqn (4 5) and the above conditions represent a constrained non-linear
optimization problem
For pulse compression sequences two characteristics are of particular concern
(Chapter 2) One is the total side lobe energy given by
E, = j:\rikf (4 9)
In a dense target environment the self clutter power at the MF output is
proportional to this quantity (Since the auto-correlation is an even function, only
one half is considered, i e the actual side lobe energy would be twice the value given
here) Another property of interest is the peak side lobe level, maxk|r(k)|, which represents
a source of mutual interference that can obscure weaker targets Therefore, it is required
51
TABLE 4.1
Binary Sequences Obtained with Various Code Initial Starting Point
Code
Lertqth
N
13
19
23
3L
37 41
43
47
53 5?
61
63
67 71
73
79
91
93
95
99
103
107 109
113
117
121
125 251 255 259
261
299 30 1
305
503
511
513
max
1 3
3
9 5
5 5
4
6 7
7
7 R 9
8
8 9
13 9
13
13
13
11 9 10
9
11 1^ i^ 17
27
18
IR
1/
25
25
23
F : 1- f k
Es U )
3.55
13.57 11.15
11.55
9.78 6.90 13.25
8.28
10.18
11.52 9.62
8.64
9.91 8.86
11.03
11.39
13.83 8.11
12.04
10.05
11.49
9.82
10.76
9.02
10.36
11.80
13.24 10.93 10.22
13.17
9.37
11.27
9 97
9.74
10.79
11.32
9.14
starting code
1
3
3
9
. 5 5 6
7
6 9
7
7
7 7
8
7 10
13
9
13
11
11 13
9
11 12
11 17 16
17
27
18
19
19
27
28
25
FUNCriONALS
Ek|r(k)|^
max
1
5
3
5
5 12 7
8
9 9 9
7 8 11
8
12 15 9
17
19
15
13
15
11
13
12
16 32
Es il)
3.55
19.11
11.15
12.38
10.08 17.13 12.39
10.46
14.88 10.60
13.06
8.64 10.54 13.31
12.76
11.97
16.19 10.52
13.91
15.36
13.49
13.29 12.54
9.24
12.29
13.32
10.92 15.08
^ max
1
3
3
4
5 5 5
5
5 8 6
7 7 9
7
8
9 8
8
9
9
9
9
9
12
10
7 13 14
13
15
17
18
16
22
22
21
E!r(k
Es (%)
3.55
11.35
11.15
16.50
11.25 15.94 16.27
14.26
12.31 16.23
10.37
8.64
15.26 12.51
11.33
16.65
15.28 10.05
13.28
10.86
15.45
12.27 15.30
14.31
13.03
13.82
10.25 12.19
12.09 12.97
12.54
16.25
11.40
13.62
13.28
14.39
14.04
1
SL,nax °f starting
code
1
3
4 4
5 6
5
4
5 8
7
7
7 9
7
8
9 8
8
9
9
9
9
9
12
11
7 13 15
13
15
17
16
16
22
23
1 21
Ekl r
SL max
1
3
3
6
5 6 6
7
7 8 8
7 7 7
9
9
12
11
11
11
11
11
11
11
10
12
12 20
/I (k)|
Es (%)
3.55
13.57
11.15
21.54
11.25 15.71 15.80
14.80
14.80
21.06 10.18
8.64
15.44 12.5?
14.71
14.73
24.21 13.58
15.95
15.56
13.49
15.84 17.29
14.79
13.69
16.12
17.22 15.41
to minimize a suitable measure whicii incorporates these characteristics Naturally, any
such measure is to some extent arbitrary, but in general will be of the form F(|r(k)|)
Specifying the desired ACF as being zero (k ^ 0), the objective may be redefined, [67] as
FAX) = ILnkMhi" (4 10)
Where p > 1 and w(k) > 0 The weighting sequence w(k) allows control of the side
lobe structure of the resulting ACF A good compromise is to set w(k) = 1 for all k, if no,
or little, prior information about the radar environment is known At this point the value
of p remains to be chosen It should be noted that for p = 2 minimization of
F,{x)=Y\r{kf (4 11)
results in minimizing the side lobe energy Es However, for p = 2 and sequence length N
> 20 the measure reacts weakly to large peak values, while for large p it will not respond
to the energy criterion, since for well behaved functions
>«"«„ .„{XV(A)1")' = maxJr(A)| (4 12)
The measure (Fp)"'' are called Chebyshev or Uniform Norms because of the
consequences of Eqn (4 12) Minimization with respect to (Fp)''*'for large p is often
reflfered to as minimax approximation, [59] In other words, the least p-th
approximation tends to the minimax approximation as p->oo Hence as suggested by
Eqn (4 12) a Chebyshev solution could be approached in principle, by successively
minimizing Fp each time incrementing the index p, i e p = 2, 4, 6, etc In most cases
acceptable minimax approximations can be obtained with relatively moderate values of p
(p=10)
Incidentally certain optimization problems can only satisfactorily be
described using several error criterion A suitable overall performance index could then
consists of a linear combination of functions of the form (4 10) i e
F - ?Li Fp<" + \2 Fp<'> + ?i3 F ; ' ' +
Where the A,, is weighting factors, would be given values, according to the importance
of, F/'>, V^, F/^>, etc
52
It is not clear which characterizes the 'goodness' of a sequence more fully, the
maximum peak value or the energy criterion Therefore, for short sequences (N < 20) p
= 2 should be adequate, while for longer sequences a larger value for p, for example, p =
4 lias tiie desirable cfTcct of reducing large single sidelobe peaks
4.2.2 Synthesis Using Element Complementation
Probably the most obvious method of synthesizing binary sequences is to
choose a number of sequences, perhaps at random, and to observe their ACF's. This
method can be improved significantly by adopting a search strategy which, starting
from an initial code, produces a succession of progressively better codes This involves
minimization of a measure of the sidelobes such as that given Eq (4 11) over a set of
discrete points in multidimensional space To be more specific, the problem is to
minimize a fiinction of N discrete variables over a set, S, of 2^ points in N-dimensional
space. Any attempt to compute all possible fianctional values becomes unfeasible, even for
moderately large N (N > 20), as 2* increases exponentially Hence, the search will have
toberestricted to asubset SI of S ie S lcS In addition iterative method must take the
discrete nature of the variables into account and, moreover, must be economical with
respect to the volume of computations
A simple search strategy is to take the current sequence, changing one of its
elements to either +1 or -1 , and to evaluate the ACF, [54] If the measure of the
sidelobes is reduced, the modification is retained and the new code is subjected to fiirther
modifications This is done iteratively until changes of the elements do not yield a
further reduction in the objective function Such a method is known as the element
complementation technique, [11], [74]
A search program based on the element complementation method was written in
fortran to find sequences with minimum side lobe peaks The initial starting points were
randomly chosen sequences In addition the sequences were also tested to see if cyclic
shifts would improve the objective fiinction, [82], [83], [84], [85], since such an
operation results in merely removing and adding bits of the sequence
53
Table 4 J shows the binary sequences obtained when optimizing sequences,
which has been generated using various methods such as maximal length sequences or
Vakman, [11] sequences with respect to the criteria XkC*)! .S^kC^)! .X^C^)! ik ft *
,ZA|r(A:)f
As expected minimization of F4 measure yields better resuhs particularly for large
compression codes. The S L energy ratio E, varies between 10% and 16% of the main
lobe energy, which gives a r m s sidelobe of approximately 0 SVN - 0 4VN AS
mentioned previously a non-negative weighting of the objective function enables control
of the sidelobe distribution along the delay axes of the ACF A designer may feel that it is
worthwhile reducing the sidelobes which are further away from the main peak, at the
expense of an increase in the close-in side lobes, by minimizing a functional of the form,
l i A - l
[67], [69]. F^ = ik\r(k)\'' This is illustrated in Fig 4 2
On the other hand, where a binary sequence is used for resolving closely
spaced targets or observing missiles in the presence of tank fragments or decoys for
example, it is desirable to have low residues near the main lobe, [26], [49] However,
ambiguities can be tolerated in range if they are sufficiently distant from the expected
target position. In this situation a weighting function of the form w(k)=N-k
j+1; if.k<N, ^^^ ^ [0; if.N,{.k<.{N-\)
Where Ni denotes the desired clear region, could be used For example
sequences of length N > 200 whose mainlobe to side lobe ratio is 200 1 or 46 dB within
10 segments length of the main peak were found Thus with a prefer choice of the
weighting function it is possible to obtain sequences with clear regions However, an
increase in the maximum and average sidelobe levels is usually observed
An extensive computer search has shown that starting with a randomly chosen
initial sequence usually results in improvement in the initial code In Table 4 2 the results
obtained with this method are compared with that of best available codes It can be seen
from Table 4 2 that results obtained are comparable with best available codes obtained
54
TABLE 4.2
Best Binary Sequences Obtained with Random Initial Starting Point
/ \
Code Length
N
13 19 23 31 37 41 43 47 59 63 67 71 79 91 95 97 99 103 105 109 115 119 123 125 251 255 259 299
Functional :
max 1r(k)| k
1 3 3 4 4 4 5 4 6 6 6 6 7 7 8 8 8 8 7 9 8 9 9 9 14 14 13 15
Elrfk)!"* k
Es (%)
3.55 11.36 11.91 10.72 11.83 12.61 12.17 8.28 14.62 11.96 12.77 9.26
16.15 16.15 13.60 13.52 15.89 14.58 13.61 12.40 14.65 15.62 11.72 15.83 14.41 14.44 14.66 15.09
Be a t known Sequences
N
k
lr(k)l
1 2 2 • 3 3 4 4 4
5 5 6
\ /
Seau-
enrp
Len
gth
7 9 Li
13
15 17
19 21
23
25
27 29 31
33
35 37 39
41
43
45
47 49
51
53
55
57
59
61
63
65 67 69
Starting sequence is
Rarknr ^egiience of
Len gth 5
added at
the
& 2
the
seauence
rlaximum
Sid e-lobe
1 o
2
2
2 2
3 3
3
4
4
4 4
4
5 5 5
6
5
6
5 6
6
6
6
6
6
6
7
6 7 7
bits are
end of
R.M.S.
Value
0.707
1.22
1.11
1.08
1.28
1.414
1.65
1.92
1.79
1.87
1.89
2.05
1.85
2.34
2.50
2.69
2.78
2.81
2.58
2.73
2.91
2.75
2.55
2.60
2.86
2.86
2.89
2.98
3.04
3.20
3.45
3.74
TAPI,E
Starting
Barker sr
Length 11
added at
4.3
sequence is
qiiPiicc of
. & 2
the
the sequence
Maximum
Side-lot
2
3 3
3
3 3
3
4
4 4
4
5 5
5
4
4
5
5 6
6
6
6
6
6
6
6
7 7 8
>e
bits are
end of
R.M.S.
Value
1.58
1.58
1.87
1.84
1.76
1.63
2.12
2.08
2.01
2.02
2.37
2.55
2.50
2.31
2.47
2.12
2.16
2.55
2.27
2.43
2.47
2.60
2.71
2.81
2.89
3.0
3.02
3.30
. 3.52
Starting sequence is
Rnr kri -.r-finciK
Length 13 & 2
'> (j(
bits are
added at tlie end of
the seauence
Maximum
Side-lobe
2 3
3
3 4
4
4 4
4
4
4 4
5
5
5
5 5 ">
5
6
6
7
7
7
7
7 7 7
R.M.S.
Value
1.28
1.41
1.77
1.81
?.12
1.82
1.80
1.9R
1.81
2.06
2.20
2.12
2.14
2.25
2.39
2.63
2.43
2.51
2.67
2.96
2.99
3.01
3.18
3.15
3.21
3.36
3.50
3.82
Seau-
ence
Length
7 9
11 13 15 17
19
21
23
25
27 29
31
33
35 37
39
41
43
45
47
49
51
53
55
57
59 61
63
65
67 69
Starting sequence is
Barker seauence of
Length 5 added at
& 2 the
the seauence
Maximum
Side-lobe
1 2 2 2 2 2 3
3
3
4
4
4 4
4 5
5
5
6
5
6
5
6
6
6
6
6 6 6
7
6 7
7
>
bits are
end of
R.M.S.
Value
0.707
1.22 1.11 1.08 1.28 1.414
1.65
1.92 1.79
1.87
1.89
2.05
1.85
2.34 2.50 2.69
2.78
2.81
2.58
2.73
2.91
2.75
2.55
2.60
2.86
2.86 2.89
2.98
3.04
3.20 3.45
3.74
TABLE
Starting
4.3
sequence is
Barker sequence of
Length 11 added at
& 2 the
the sequence
Maximum Side-lob
2 3 3
3
3 3
3
4
4
4
4
5 5
5
4
4
5
5
6
6
6
6
6
6 6
6
7 7
8
•e
bits are
end of
R.M.S.
Value
1.58 • 1.58 1.87
1.84
1.76
1.63
2.12
2.08 2.01
2.02
2.37
2.55 2.50
2.31
2.47
2.12
2.16
2.55
2.27
2.43
2.47
2.60
2.71
2.81
2.89
3.0
3.02 3.30
3.52
Starting
Barl<
sequence is
;er sequence of
Length 13
adde
the id at
& 2
the seauence
Maximum
Side-lob
2 3
3
3 4
4
4
4
4
4
4 4
5
5
5
5
5
5
5
6
6
7 7
7
7
7 7
7
ip
bits are
end of
R.M.S. Value
1.28 1.41
1.77
1.81
2.12
1.82
1.80
1.98
1.81
2.06
2.20 2.12
2.14
2.25 2.39
2.63
2.43
2.51
2.67
2.96
2.99
3.01
3.18
3.15
3.21
3.36 3.50
3.82
71
73
75
11 79 81
83
85
87 89 91 93 95 97 99
101
103
7 9
9
9 9
9 9
10
10 10 10
10 10
11 11
11
11
3.89
3.88
3.82
3.97 4.03 4.20
4.39
4.52
4.61 4.63 4.68 4.75 4.85 4.79 4.84
4.87
4.95
8
8
8
9 8 9
9
9
8 8 8 9
9 8 9
9
9
3.38
3.47
3.49
3.54 3.52 3.72 3.67
3.66
3.74 3.83
3.94 4.02 3.94
4.04 4.17
. 4.36
4.23
8
8
8
8
9 8
8 9
8
9 9 9 9
9
9 9
9
3.65
3.62
3.63
3.69
3.64 3.77
3.89 4.02
4.13
4.22 4.13 4.21 4.43
4.55
4.54 4.45
4.65
Sequ
ence
Length
11 15 19 23 27 31 35 39 43 47 51 55 59 63 6 7
71 75 79 83 87 91 95 99 103 107 111 115 119 123 127
Starting
Barker
Length are add of the
se 11 led se
Maximum
S; Lde-I
3 3 3 4 4 4 4 5 5 6 7 7 7 8 8 9 8 8 9 9 9 9
10 10 10 11 11 11 12
TABLE
sequence is quence of &
at 4 bits the end
quence
.obe R.M.S. Value
1.38 1.50 1.74 2.12 2.02 2.33 2.35 2.58 2.80 3.22 3.01 3.19 3.29 3.52 3.64 3.70 3.88 3.78 4.05 4.28 4.32 4.60 4.73 5.1 4.92 4.97 5.2 5.48 5.36
4.4
Sequence Length
13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97 101 105 109 113 117 121 125 129
Startir Baj rker Length are add of the
ig sequence is sequence of 13 led
& 4 bits at the end
sequence
Maximum S: Lde-1
3 3 3 4 4 5 5 5 6 6 6 7 7 7 8 9 9 10 10 10 10 10 10 10 11 11 11 12 12
R.M.S. obe Value
1.41 1.81 1.77 1.98 2.23 2.50 2.46 2.45 2.38 2.69 3.25 3.43 3.87 4.03 4.35 4.36 4.64 4.71 4.73 4.72 4.73 4.62 4.58 4.83 4.90 5.01 5.09 5.26 5.33
L.O
0.5
|r(T)| objective function of the form^
I Kjrtk)|*
/ -21dB -33dB
I 255T
Fi9« . 4.2 ACP of 255-element binary sequence for linear weighting sequence.
1.0 ~l |r(T)|
0.5
AxJ'^'WKJ^. A//// >u / ^ C T N7.
-22dB
T lOlT
Fig. 4.3 - ACF of optimum 101-element binary sequence with random initial starting point.
with complex mathematical operation with extensive search. The r.m.s. level is about 0.4
VN and maxk |r(k)| ranges from 0.6 VN to VN but generally does not exceed VN as
illustrated in Fig. 4.4. A typical ACF of such a sequence is depicted in Fig. 4.3.
It has been observed that there are a number of good binary sequences which
have the lowest sidelobe known for a given length. In general, however, they will differ
in sidelobe energy performance.
The element complemientation method requires a computation time which
increases almost linearly with the code length N. In most cases the optimum was found in
about 3 to 4 iterations. (An iteration consists of complementing the N bits of the
sequences plus N cychc shifts). On an VAX-11/780 digital computer_ good binary
sequences of length as large as N = 900 were found in less than 30 minutes. On a time
sharing computer a search program of this kind runs at low priority very cheaply, since
both input and output data are extremely small.
4.2.3Synthesis using Cyclic Shift and Bit Addition
In this section a different approach of minimizing the objective function FP
based on cyclic shifting, [82], [83], [84] ,[85] of a particular sequence is discussed. A
sequence is taken randomly and all its cyclic shifts are tested with respect to Max|r(k)|
and S.L. energy ratio E,. The sequence for which maximum side lobe and S.L. energy
ratio is minimum is retained.
This is the optimum sequence for that length. To increase the dimension or
length of the sequence 2-bits or 4-bits are added at the end of the optimum sequence of
length n obtained by shifting procedure. Therefore, the new sequence generated would
be of length N = n + 4 o r N = N + 2, while adding the additional bits all the
combinations are tried. In the 4-bit case, there would be 2'*= 16 combination as
against 2^=4 combination in the case of 2-bits addition. Each sequence formed by
increasing the length from n to N, is shifted cyclically and the ACF of each shift is
evaluated. The sequence for which the MSL and RMS is minimum is taken as the
optimum sequence. The procedure is repeated till the desired length of sequence is
obtained. The resuUs are shown in Table 4.3 & 4.4. In this method the initial or
55
MAX. SIDE LOBE
• ^
n
I s
fe« t
i I
starting sequence is the Barker sequence The Table 4.4 shows the results when the
starting sequence is the Barker Sequence of length 11 and 13 respectively and 4-bits are
added at the end of each sequence Table 4 3 shows the results when the starting
sequences are the Barker Sequences of length 5, 11 and 13 respectively and 2-bits are
added at the end of each sequence. As can be seen from these tables, the sequences
obtained in Table 4.3, where only 2-bits are added at the end of each sequence, are better
in most of the cases in comparison to the sequences obtained in Table 4 4, where 4-bits
are added at the end of each sequence The time and computation involved are also less
in the case of sequences given in Table 4.3, because of less number of combination (2^
= 4 combination only in comparison to 2'*= 16 combination). Figure 4.4 shows
the variation of MSL with sequence length As can be seen from the figure MSL is
below the upper bound specified by Vn in all the cases. The comparison of results
obtained in Table 4 3 with those given in Table 4 2, clearly shows that, the results
obtained in the proposed method are comparable to those given in Table 4 2 and in some
cases there is improvement It is also observed that the computer time taken by this
method is also lesser than the methods given in Table 4.1 & 4 2 This is because of less
number of computation involved. Fig. 4 5 shows the variation of maximum sidelobe
level with sequence length. It can be seen from the figure that the maximum sidelobe
levels remain within the limit VN for each sequence length 11
4.3 UNIFORM SEQUENCES WITH LOW AUTOCORRELATION SIDELOBES AND SMALL CROSSCORELATION
Much attention has been paid by many authors to the construction of binary
sequences having ACF's as small as possible away from the coincidence peak
However, little is known about sequences with small cross-correlation Such sequences
have many practical applications For example, they may be used as address codes in a
time division multiple-access (TDMA) system, where information from several data
sources is to be transmitted over a channel, [61], [86], [89]
56
It has been shown that for most good binary sequences of length N (N>13),
the attainable sidelobe levels are approximately VN The mutual cross-correlation
peaks, however, of sequences of the same length tend to be much larger and are
usually in the order of 2VN to 3 VN Consequently, the objective in this section is to find
set of binary sequences of length N with auto-correlation side lobes and cross-correlation
peak values both of approximately VN
4.3.1 Statement of the Problem
The more complicated problem of finding a set of uniform pulse compression
codes which besides having small ACF sidelobes also have small cross-correlation, can
be approached using optimization techniques The ACF and cross-correlation of L
arbitrary sequences of length N are given by
N-I- |* |
rAk)= Z«(«).«*(« + *) (4 13) H 0
rAk)= Y.b(n).b'(n + k) (4 14) n=0
N-l-\k\
r,{k)= Tc{n).c'(n + k) (4 15) «=0
r j * ) = Z/(«) . / ' ( /« + *) (4 16) 'L n 0
AT-l-likl
ri2(A)= Y.a{n).b'{n + k) (4 17) « = 0
N-l-|ft|
r„(/^)= T,h{n).c'{n + k) (4 18)
N-l-\ki
r,Ak)= llain).c'in + k) (4 19) «=o
Wherek = 0, ±l ,±2, ,± (N- l )
57
TABLE 4.5
Correlntjon Properties Obtained IJBinff Numerical Minimization
iSequ-
|ence {Length
! N
! 19
1 101
1 109
1 19
; 101
1 109
Peak Side Lobe<
iiiax|(rj(k)!!«axjr,(k)|
k ! k
3 1 3
4 1 4
5 ! 4
5 1 5
5 i 6
7 ! 6
7 1 7
7 ! 6
7 1 8
8 ; 8
9 1 8
11 I 9
10 1 9
-• 1 I • 1 1
3 ; 4
4 I 5
6 : 7
7 ! 6
6 ! 7
8 1 9
9 1 10
9 1 10
10 ; 11
9 1 10
11 1 10
9 1 11
13 1 12
•ax|rj(k)|
k
8 10
10
9
4
5 7
6
7
9
10
9
11 10
13
12 12
Peak Cross Cot
•ax|rj2(k)|
k
8 10
10
13
13
22 18 18
20 19
18
24
26
5
7 8
9
9
10 10
11
12
12
13
15 14
«ax!r23(k)!
k
7
10
10
12
12
18
19 20
19 20
19
23
24
5
8 8
10
9
9
10
11
12
12
13
14 14
r.
maxjr.jfk)
k
8 10
10
12
13
15 17 19
21 19
20
21
25
6
7 9
9
10
9
11
10
11
11
13
14
15
*1
11.45
16.5
11.50
16.12
19.2
12.5
11.3
12.20
14.66
16.68
16.25
12.56
14.70
20.56
16.65
27.03
27.89
21.47
31.54
29.89
26.24
27.17
25.59
24.36
25.83
29.89
Side Lobe E
2
11.25
10.72
11.82
15.52
12.17
15.88
12.65
12.59
14.56
14.56
16.65
14.59
13.51
22.36
18.62
31.23
2". 74
21.90
28.64
28.86
27.89
29.59
24.87
28.18
30.23
28.51
3
11.45
11.82
11.95
14.85
14.5
16.55
15.45
14.65
15.45
15.56
15.89
15.69
14.98
23.62
22.35
30.59
30.36
22.54
29.54
29.65
29.65
30.01
29.72
27.53
29.81
27.59
nergv Ratio
^2
105 97.5
97.11
97.2
107.2
107.41
108.60
102,27
98.47
94.81
97.51
98.61
105.71
68.32
87.47
70.87
61.45
77.59
67.25
66.75
74.83
67.5
76.56
67.35
71.35
75.35
«23
108
98.8
99.2
98.5
105.6
106.52
107.52
105.65
99.56
98.6
99.62
97.59
106.89
71.23
91.56
73.23
65.45
78.81
69.54
68.54
76.51
71.85
75.54
69.54
73.52
72.32
13
109
100.2
98.8
99.65
104.4
104.9
106.95
104.59
100.23
99.45
97.56
99.431
107.32
69.89
89.73
71.34
64.34
76.54
70.13
71.59
78.89
69.71
80.IS
68.13
69.8"
76.5«
In the binary case, sequences which resuh in the best possible auto-correlation
and cross-correlation must satisfy the conditions
iO: (N - k).is.even
± 1 ; iN-k)M.oJ<l <^^'"
The above conditions form a system of non-hnear equations in the unknown a(l),
a(2), , a(N-l), b(l), b(2), , b(N-l), 1(1), 1(2), , 1(N-1) It can be easily observe
that it is impossible to satisfy (4 18) simultaneously, except for the two trivial cases of
N=l, 2 Again an approximate solution to this set of equations is sought using numerical
methods
As mentioned previously one of th emost important steps in the optimization of
any design or process is the choice of the optimization criterion For pulse
compression sequences the properties of concern are the total sidelobe energy and the
peak side lobe For a set of sequences, however, an additional important criterion is
the peak magnitude and energy of their mutual cross-correlation function This may be
of concern in satellite communication systems such as TDMA where the problem of
unique word synchronization requires sequences which not only have good auto
correlation properties, but also should have as little cross correlation as possible, [61],
[86]
Therefore, what is required is the minimization of a suitable measure which
characterizes 'goodness' of a pair of sequences Any such measure is to some extent
arbitrary but in general will be of the form F {|r(k)|} For any particular set of sequences
the performance index for each individual correlation function is defined as
F, = l | r , (A)f (4 21)
F, = 'f\r,(k)\" (4 22)
F, = Z\rM)\* (4 23)
58
N - 1
F,^= Z k(Ar)f (4 24) A — ( N - l )
N-\ z ik (N I)
^23= Z ' k„(A)r (4 25)
Again only one half of the ACF is considered, since it is an even function The
optimum sequences a(n), b(n), c(n) etc are determined from the condition
Fi + F2 + + FL = min
and (4 26)
F12 + Fi3 + F23 = min
This can be accomplished by minimizing a linear combination of the performance
indices
min F = XiF, + X2F2 + + >.LFL + W F12 + ^2'Fi3 + ^3^23 (4 27)
Where A,, and X,' are weighting factors chosen according to importance of Fi, F2, FL and
F12, Fi3, F23 etc
The problem of minimizing F is one of minimizing of 2(N-1) discrete variables,
which can assume only two values ±1, over a set of 2^"'^ points In addition a train of
values of weighting parameters are required If X,' >X^ minimization results in sequences
with good cross-correlation but poor auto-correlation For small A,,, and X,'=0
sequences with good ACF's are obtained
The choice for the best value of A,,, and X,' depend on the specific applications as
well as the knowledge of the behaviour of the functional F, including the interaction,
between the performance indices Obviously Fi, F2, FL etc are independent of each
other This is, however, not the case for F12, Fi^, F23 etc
4.3.2 Results of the Synthesis
59
The element complementation method is used to minimize the functional F of the
form
F = F, + Fz + F, + F,2 + Fi3 + F23 (4 28)
Taking a set of three sequences a(n), b(n) & c(n) functional F from Eqns 4 13-
to- 4 18 is given by
N'l I 14
*=1 A=l A=l * = - ( N - l ) *=- (JV- l )
(4 Zy)
A=-(Ar- i )
For the simplest case, A,„=A,,'=1 is chosen initially There are number of ways of
applying the optimization procedure One approach is first to change all the elements of
sequence a(n), keeping b(n) & c(n) fixed that is minimization of
/^ = / l + Z h ( A ) f + Z k ( A ) r + Z |r.3(A)f (4 30)
is carried out, where
^=Z'{|r,(A)f+|r3(A)f}+ Z' \r^{k)\* (431) A=l A = - ( N - l )
is now a constant term The next step is to repeat the process for sequence b(n) while
a(n) & c(n) remain unchanged, i e
F = B+T.\rAk)\'+ Z' {k(A)P+|r,3(A)f} (4 32) A - l ft = - ( N 1)
60
Ijj Sequence a (n) s
4 - - + * + •••• • • - + • • - • » • + -• • - * • - • - » . - •
. + + - • - • • • - - + - • -• - • - + • • I -t ' » - t ' - f - i - - + - - -- • • • • ••• + • • - - + •»•-% •
•
l .O
| r ^ ( T ) |
W^^iVi^ - low
.vS' MvvAVwyV
Ci) Sequence b (n) «
• - - • - • • « • • • • • - - f . . - ! . .
• • • 4 - . 4 . 4 . . - • • - + . 4 . 4 . 4
• • - • - • • - • • 4 • « • • • • -• 4 4 4
- - • - • - • • • I - * • • • + • 4 * - l > - -
' • •" I - : It) I
iMl&^M t ^
- l O l T lOXT
10 1 . 0 -
k j t t ) |
- lOlT »IT O lOlT
FisJ. 46 Correlation functions of two binary sequences
of length N - lOli (a) maxlr. ()c)| - -21 dB
k * , (b) roax|rj(k)| - -19 <»
k .(c) naxlr (k) | - -18 da
where
B=i:{\r,{k)\\\r,{k)[}+ t \r,Ak)[ (4.33) A=l ft=-(N-l)
is a constant term.
The next step is to repeat the process for sequence c(n) while a(n) & b(n)
remain unchanged. This is done iteratively until (local) minimum is reached.
An extensive computer search was carried out using the element
complementation techniques. The search was started with a randomly chosen set of binary
sequences or sequences generated in section 4.2.3. Some of the results are smmarized in
Table 4.5 for various values of N, where ei, 62 and en are the normalized side lobe energy
ratios given by
ei.2,3=10'E,.2,3/N'
e,2 = 10 E,2/N^ e,3 = 10 E.j/N^ e23= 10^ £23/^^
For X,i'=0, the rms , and peak values of the auto-correlation sidelobes are around
O.45VN and vN respectively. These values correspond to those obtained in section
4.2.2 and 4.2.3. It should be noted that the cross-correlation energy for this case is
approximately N^. The results in Table 4.5 show quite clearly that sequences having
optimum ACF do not give smallest cross-correlation values and thus appear to be
correlated. Table 4.5 also shows good agreement with the sought after energy
distribution when a measure given by Eqn. 4.29 is minimized. Moreover, the cross-
correlation peak values have decreased considerably to approximately 1.4'vN as
compared to 2" N to 3VN for. However, as expected, this improvement is achieved at the
expense of an increase of the maximum auto-correlation side lobes which are, except, for
small values of N, usually of the same order. In Fig. 4.6 a representative graph of two
sequences of length N = 101 and the magnitude of their correlation functions is
sliown In all cases the miainiuin cross-correlalion peak values obtained have been
found to be greater than the minimum bound estimated as shown in Fig. 4.7. For X,3
=0 and X,2'=A,3'=0 the synthesis will give pair of sequences with good auto and cross-
correlation properties.
61
It has been shown that binary sequences whose largest auto-correlation side lobes
and cross-correlation do not exceed unity do not exist for N>2. However, adopting a
numerical optimization technique it is possible to find sequences which are satisfactory for
most practical applications With a proper choice of A, the weighting parameter,
a significant improvement of the cross-correlation peak value can be achieved at the
expense of only a relatively small increase in peak side lobe level. It can be observed that
minimization merely results in a redistribution of the energies contained in both the auto
correlation and cross-correlation functions
4.4 TERNARY SEQUENCES
The performance of a binary sequence can be improved significantly using
numerical technique The attainable energy ratios and peak side lobes are of the order
of 20% and VN respectively The still relatively large residues might be the limiting
factor for particular applications such as precision tracker^ One approach to try and
improve the range resolution and yet to maintain the binary nature of the sequence is
to introduce zeros, that is by setting a number of elements in the sequence equal to
zeros The resulting sequence can be regarded as having three possible levels, namely ±1
or 0, and is sometimes referred to as interrupted binary sequence or ternary code
The loss in transmitted signal energy associated with ternary codes can be kept
small provided that the number of zeros L, is small compared to the code length N For
L = 0 1 N, a reduction of 0 4 dB in SNR is obtained If L is large (L = 3N/4) the code
becomes energy inefficient and approaches the staggered pulse train properties, [22]
Hence the performance of ternary code is somewhere between these two extremes,
depending on the number of zeros, L
The question which arises now is how much can the performance of a
code be improved by allowing the sequence elements to take on values +1, 07 For an
estimation of the rm s side lobes level consider a random sequence c(n) whose elements
62
can assume the values -1 , 0, +1, with probabilities p(-l), p(0), p(l) Assuming a
symmetric probability distribution, that is
p(-l) = p(l) = w
and
p(0) = 1 - 2w
The ACF is given by
r(k)= Y,c{n).c{n + k) (4 34) «=0
The elements c(n) are independent random variables Making the substitution Nk
- N-k, Eqn (4 34) can be written as,
/•(*)= Z \ ( / i ) . (4 35) n 0
Where q(n)=c(n)c(n+k) is a random variable taking on values -1 , 0, 1 with probabilities w,
l-2w, w Each q(n) is independent (k ^ 0) with zero mean and variance a^ = 2w The
probability distribution of the sum of two random variables is the convolution of their
individual distribution Having Nk terms in (4 35) this operation as to be performed Nk
times For Nk reasonably large, the central limit theorem applies, [44] The distribution
will be gaussian with zero mean and variance Ok , where Ok is simply the sum of the
variances a^, that is
= 2w(N-k)
and
P^k) = (2/7ia')'^'exp[-r\k)/20k^] (4 36)
where
Oo = (2wN)^
63
The r m s value of kth side lobe is
ak = Oo(l-IC^)'/'
It is clear that the r m s value decreases with distance from the main peak
Ok = Oo for k small
Ok = V(2w) for k large (k < (N-1)) (4 37)
For strictly bipolar case w =1/2 and Oo = VN, which is familiar result To
obtain small r m s side lobe values Eqn (4 36) indicates that w should be small
However, the smaller with greater the loss in transmitted signal energy For w = 1/4 the
decrease in SNR is about 3 dB The r m s side lobe on the other hand will be reduced
to 0 7N This is not so significantly better in comparison to VN for the strictly binary
case and in general to obtain low side lobes one must be prepared to introduce
more than N/2 zeros However, it is shown that with a proper choice of the zero positions
some improvement in peak side lobe is obtained with little loss in energy performance
The problem is to find the optimum zero positions which yield the maximum
reduction in side lobe energy and side lobe level The search procedure can be carried
out in a similar manner as described in section 4 2 2 by a simple modification of the
basic element complementation method
The results are given in Table 4 6 for a relatively small number of zeros In many
cases the peak side lobe is reduced by three units while suffering a loss in SNR of less
than 1 dB For further improvements more zeros have to be introduced at the expense
of the energy performance of the code
An interesting feature of ternary codes is their use in a multiplex pulse-
compression system, [82] If one of the zero elements of a ternary code, C, is changed to
+ 1 or -1 resulting in the sequences C+ and C- respectively, it can easily be seen from
Eqn (4 34) that the coherent summation of their individual ACF's is given by
r.(k)=l/2{r..(k) + re.(k)} (4 38)
64
Performance of Ternary Sequence (L denotes number of zero-elements)
/ \
Code Length
N
19
23
31
37
41
43
47
53
59
61
63
67
101
103
105
107
119
121
125
251
Binary Sequences
max 1r(k)| k
3
3
4
5
6
5
5
5
8
7
7
6
9
9
9
9
9
10
7
13
Ex (%)
11.36
11.5
16.55
11.25
16.66
16.28
14.26
12.32
16.23
13.81
8.64
14.44
15.82
15.45
13.68
12.27
13.80
13.82
10.25
10.10
L
1
3
2
2
5
6
8
4
8
6
2
3
9
10
10
7
7
6
5
9
Ternary
max 1r(k)| k
2
2
3
3
3
3
3
4
4
4
5
5
6
6
5
6
7
7
6
11
Sequence
Es (%)
7.72
6.00
10.46
8.25
9.57
6.94
3.94
9.50
11.24
8.51
9.03
12.65
11.32
9.74
9.14
10.20
10.01
12.26
6.67
8.64
Loss (dB)
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.24
0.61
0.31
0.25
0.57
0.77
0.81
0.34
0.63
0.45
0.14
0.20
0.41
0.44
0.44
0.29
0.30
0.26
0.25
0.29
\ /
For a ternary code with L zeros, 2^ sequences with this property can be found.
4.5 SUMMARY
In this chapter the application of numerical methods to the design of discrete
coded waveform (pulse trains) has been investigated. The objective was to study the
range resolution and clutter rejection performance of these discrete coded waveforms.
Sequences which optimize the correlation properties, defined by suitable cost
functionals, were considered. Without prior information about the radar environment
the choice of such a measure of 'best' is to some extent arbitrary. It has been shown that
minimizing an F4-measure has the desirable effect of reducing the peak sidelobes as
well as the side lobe energy. However, such a measure is highly non-linear and
multimodel and at present there is no criterion to tell whether the obtained extremum is
a global minimum.
The design of binary sequences using cyclic shifting and bit addition also yields
sequences with good auto-correlation function properties. These sequences can also be
used as a starting point for the synthesis of pairs or set of binary sequences having low
auto-correlation side lobes and small mutual cross-correlation.
Finally, set of phase coded sequences having low auto-correlation side lobes
and small mutual cross-correlation have been designed. These sequences are particularly
useful in spread spectrum multiple-access systems. It has been shown, however, that
improved cross-corrclalion properties can only be obtained at the expense of an increase
in the auto-correlation side lobes.
65
Although, numerical methods have proved to be successful they are not without
their weaknesses. From the theoretical point of view the most serious objection is that the
results are not unique. Thus for a given design objective, the sequence to which the
procedure converges is dependent on the initial choice of the starting sequence. Thus each
individual applications the initial point would have to be chosen judiciously. Furthermore,
when very large pulse trains with hundreds or thousands of pulses are required
difficulties of a computational nature arise. These difficulties can be overcome to some
extent by trying to generate longer sequences by combining short ones, [15].
Unfortunately, the ACF's of good short sequences do not reveal any pattern which
would suggest a plausible rule for their construction. Although various techniques to
obtain combination codes exist, [15], little efforts has been made to found
construction method which diminishes maxk|r(lc)|.
66
CHAPTEE-
AMPLITUDE AND PHASE MODULATED PULSE TRAINS
66
5.1 INTRODUCTION
Most modern high performance radars use traveling-wave tube amplifiers to
obtain coherent transmission As pointed out previously these tubes work most
efliciently under constant amplitude conditions Moreover, good amplitude modulation
(AM) is difficult (and very expensive) to achieve with these devices Therefore, only
purely phase modulated pulse trains have been considered so far However, with the
emergence of solid state microwave sources, efforts are being made to replace the
relatively large and expensive vacuum devices by low power solid state elements and the
waveguide elements by planar circuits With these new components the size and costs
are reduced and anumber of commercial applications become feasible
The contribution of solid-state devices has also been significant in areas where
performance was previously inadequate For example, it is much easier to use any form of
modulation with solid state components Although, AM is not an efficient method to
provide the large time-band width required for good radar performance, it does provide
a excellent means of improving the resolution capability Consequently, this chapter
treats the problem of finding energy efficient amplitude and phase modulated (a m ph m)
or multilevel pulse train
N-\k\
r(A)= Z a(n)a'(n + k)^ n 0
5.2 HUFFMAN SEQUENCES
Huffman, [20] has shown that it is possible to derive sequences a(n) of any
arbitrary length (N+1) whose ACFs have the property
£ ; Jt = 0
0; \k\^Q,N (5 1)
r(Ny,...\k\^ N
Where E is the energy of the sequence
This ACF is zero for all time shifts excepts for the unavoidable end sidelobes
Without loss of generality the end side lobes r(N) can be set to unity Using the familiar ZT
notation, the ACF of a Huffman code reduces to
R(Z) = Z"" A(Z) A*( 1 /Z) = 1 + EZ-^ + Z^^ (5 2)
67
where
^(Z)=ifl(/ i)Z " = a ( 0 ) n ( l - Z ' .ZJ n=0 i=l
For a pulse train A(Z) to have the property (5.1) Huffman, [20] showed that
the roots Zi of A(Z) must lie at equal angular intervals (2TC/N) in the complex Z-plane
on either of two origin centered circles whose radii are given by
X = (E/2 + [(E/2)^-lf'y'^
l/X = (E/2-[(E/2)^-l]'/0'^ (5.3)
Since the polynomial A(Z) has N roots, there are 2^ possible root patterns and thus
2 ^ Huffman codes with the same ACF that can be derived for a given Energy E and
sequence length (N+1). Although some inefficiency in the use of transmitter power
may be acceptable in order to obtain the impulse like property (5.1), it would certainly
be wasteful not to seek the most efficient sequence for an application. A figure of
merit for the energy distribution of a sequence is the energy ratio or energy
efficiency defined by
E.R = E/{(N) max„ |a(n)|^} < 1 (5.4)
The maximum value of E.R. is attained for purely phase modulated pulse trains
such as binary sequences. The energy ratio depends on two independent variables,
namely, the total energy E of the coded waveform and the magnitude of the largest
coefficient ofA(Z) or the largest pulse of the code, denoted by max„|a(n)|. By referring
to eqn. (5.3), it can be easily verified that the energy
E is given by
E = X'' + X"'' (5.5)
and thus is a junction of the radius X only. The maximum amplitude max„|a(n)|, however,
depends on the particular choice of the N zeros for A(Z) as well as the radius.
Unfortunately, a mathematical method has not been found which leads, without trial and
error, to the most efficient Huffman code.
68
The design of a Huffman sequence in general requires the choice of the code
length (N+1), the circle radius X and the zero pattern for which the magnitude sequence
|a(n)| is most uniformly distributed. This, however, remains an unsolved problem. Direct
evaluation of all 2 ^ possible root patterns, for a given radius X and N, is not feasible if N
is large (N > 20). This remains so even if account is taken of the zero patterns formed
from others by rotation in the Z-plane through an angle <}> or by other
transformations for which the energy ratio is invariant, [74]. A general trial and error
procedure is to choose a root pattern, perhaps at random, and to compute the sequence
for a succession of values for the radius X. Other methods devoted to this problem have
been suggested by Ackroyd, [75], using the stationary phase principle, which is
applicable for an arbitrary length sequence. Another method which immediately comes
to mind is a random solution of the zero pattern. In fact Huffman suggested that in
order to maximize the energy ratio, the roots should be chosen in a random fashion with
approximately half the zeros on each circle. He then concluded further that the
optimum energy ratio should be proportional to (N+l)'"' ^. In general correlation
properties are used to judge the randomness of a sequence, that is a sequence is
uncorrected with itself, i.e. random, if its ACF has uniformly low side lobes.
Although various methods described above may be acceptable for designing
Huffman codes of moderately large length (N=100), major difficulty arise for longer
sequences because of difficulty of the computational nature due to factorizing
polynomials of degrees larger than 100. Following sections describe some useful methods
for designing the multilevel sequences.
5.3 DESIGN BASED ON OPTIMIZATION
The numerical minimization method discussed in Chapter 4 (for the design of
binary sequences) is extended here for the design of multi-level (or a.m.ph.m.)
sequences. The objective function or optimization criterion is the same as presented in
section 4.3 and 4.5. However, for multi-level sequences where the sequence can have any
number of levels, the choice of starting point becomes difficult as the no. of levels
increasing. The minimization algorithm searches the minimum with respect to phases
69
TABLE 5.1
SEQUENCES OF HUFFMAN CODES
SEQUENCE LENGTH
4
5
7
11
13
SEQUENCE
-1.00 1.00 0.618 0.618
1.00. . -1.00 0.50 1.00 1.00
-0.50 1.00
-1.00 0.00 1.00 1.00 0.50
0.353 -0.336 0.161 1.000
-0.989 0.922 0.989 1.000
-0.161 -0.336 -0.353
1.00,0 1.093 0.597 0.791 0.686 -0.905 -1.086 0.905 0.686
-0.791 0.597 •
-1.093 1.000
ENERGY EFFICIENCY
0.691
1
0.850
0.643
0.485
0.778
AUTO CORRELATION FUNCTION
1.000 0.000 0.000 -0.224
1.000 0.000
oiooo 0.000 0.235
1.000 0.000 0.000 0.000 0.000 0.000
-0.056
1.000 0.. 000 0;000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
-0.023
1.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.099
MAXIMUM SIDELOBE LEVEL
0.224
0.235
0.056
0.023
0.099
SIDELOBE ENERGY
0.0999
0.1107
0.0062
0.0011
0.0196
TABLE 5.2
Sequence derived from windov; function for a=2 Optimization criterion is sidelobe energy
SEQUENCE LENGTH
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43 •
45
47
49
ENERGY RATIO
0.81
0.77
0.75
0.73
0.72
0.71
0.71
0.70
0.70
0.70
0.69
0.69
0.69
0.69
0.^9
0.68
0.68
0.68
0.68
0.68
0.68
0.68
0.68
SIDELOBE ENERGY
0.052
0.214
0.242
Q.261
0.248
0.188
0.219
0.169
0.058
0.152
0.268
0.226
0.167
0.224
0.195
0.152
0.167
0.175
0.182
0.229
0.221
0.294
0.233
MAXIMUM SIDELOBE LEVEL
0.154
0.246
0.289
0.284
0.228
0.237
0.185
0.155
0.119
0.162
0.195
0.219
0.144
0.127
0.167
0.097
0.116
0.134
0.129
0.160
0.137
0.136
0.125
TABLE 5.3 .
Sequence derived from window function for a=3 Optimization c r i t e r i o n i s sidelobe energy
SEQUENCE LENGTH
5
7
9
11
13
15
17
19
21
23
25
27
1 1 i 35
37
39
41
43
45
47
49
ENERGY RATIO
0.68
0.64
0,61
0.60
0.59
0.58
0.58
' 0.57
0.57
0.56
0.56
0.56
0.56
L 0.56
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
SIDELOBE ENERGY
0.002
0.074
0.161
0.227
0.206
0.194
0.081
0.154
0.148
0.175
0.194
0.196
0.184
0.182
0.201
0.184
0.184
0.191
0.178
0.166
0.186
0.154
0.166
MAXIMUM SIDELOBE LEVEL
0.025
0.148
0.189
0.281
0.181
0.209
0.154
0.135
0.145
0.139
0.167
0.175
0.155
0.143
0.150
0.169
0.153
0.157
0.143
0.112
0.133
0.091
0.120
TABLE 5.4
Sequence derived from window function for a=2 Optimization c r i t e r i on i s "max. sidelobe level and sidelobe energy"
SEQUENCE LENGTH
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
ENERGY RATIO
0.81
0.77
0'.75
0.73
0.72
0.71
0.71
0.70
0.70
0.70
0.69
0.69
0.69
0.69
0.69
0.68
0.68
0.68
0.68
0.68
0.68
0.68
0.68
SIDELOBE ENERGY
0.052
0.214
0.209
0.261
0.199
0.230
0.133
0.301
0.353
0.184
0.358
0.175
0.157
0.213
0.309
0.268
0.288
,0.317
0.218
0.205
0.276
0.234
0.213
MAXIMUM SIDELOBE LEVEL
0.154
0.246
0.213
0.284
0.154
0.169
0.142
0.169
0.177
0.142
0.166
0.132
0.100
0.135
0.146
0.147
0.150
0.137
0.125
0.112
0.135
0.108
0.106
TABLE 5.5
Sequence derived from window function for a=3 Optimization c r i t e r i o n i s "max. sidelobe l eve l and sidelobe energy"
SEQUENCE LENGTH
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
ENERGY RATIO
0.68
0.64
6.61
0.60
0.59
0.58
. 0.58
0.57
0.57
0.56
0.56
0.56
0.56
0.56
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
0.55
SIDELOBE ENERGY
0.002
0.074
0.161
0.302
0.206
0.213
0.092
0.135
0.148
0.161
0.136
0.212
• 0.193
0.256
0.237
0.200
0.335
0.284
0.282
0.225
0.216
0.213
0.209
MAXIMUM SIDELOBE LEVEL
0.025 •
0.148
0.189
0.219
0.181
0.169
0.135
0.117
0.145
0.121
0.123
0.136
0.125
0.145
0.118
0.126
0.173
0.155
0.134
0.114
0.104
0.126
0.096
Energy natlo
0,0'
o.e
0.4
0.2
-&'—£>, A - A — A — A — A — A A A -
0> L. J I I I I I •I. I I I L
6 7 e 11 13 18 17 19 21 23 26 27 29 31 33 36 37 39 41 43 46 47 49
Soquenoe Lenoth
-— Alpha • 3 - ^ Alpha - 2
' ^ S . I Enoryy Rollo v /» Scqucnoo Length
The Alpha in figures i^ denoted by "a" in the text
magnltuda
0.8
0.8
0.7
o.e
0.5
0.4
20 25 30 •equence length
—~ •Ideloba energy —j—max. •Ideiobe
Fig. 5.2 Autocorrelation Function v/« Time (•lpha-3)
45 50
magnltuda
0.0
0.8
0.7
O.e
0.5
0.4
0.3
0.2
0.1
10 IS 20 25 30 taquartce length
3 5 40 45 50
—^ aldolobo energy - ^ m a x . aldolobe
ng.5.3 Autooorretatlon Function v/a Time (•)ph«-2)
magnltuda
o.e
o.e
0.7
o.a
0.5
0.4
20 25 30 •equenca length
35 40 45 SO
' aldaloba energy 'max. aldaloba
fig. 5.4 Autooorrelallon Funollon v/a Tlma (alpha-3)
moonltudo
0.0
0.8
0.7
O.e
0.5
0.4
10 15 20 26 30 aoquBnoe length
35 40 45 SO
~*-aldolobe energy 'max. aldoloba
Fig. 5.5 Autooorrelallon Function v/a Time (alpha>2)
0.4
0.3
0.2
0.1
O
AinpKtudo
If *
1
^
1
X
•
1
X
/ • 1
A
1
A
X ><
^CJ \ ?^^^^^]j>*;;Ar -^
X V-'' ' '^
1 1 1
•
\ ^
1
10 15 20 25 30 Sequence length
35
~*-aldelobe energy - ^ a l d e l o b e energy ^ ^ a l d e l o b e energy
ng .S . ( Sidelobe energy va Sequence length
40 45 60
' aldelobe energy
Amplitude
0 5 10 15 20 25 30 Sequence lohgth
-alpha-! "S-olpha«S -^a lpha-Z -S-i |pha-3
fig. 5.7 Max. aldelobo va Soquonoo length
amplitude 1
0.0
0.8
0.7
0.8
0.5
0.4
0.3
0.2 -
0,1 -
10
•N-5
Fig. 5.8(ii) Autooorralotlon Funatlon va TIma
amplitude
0.0
0.8
0.7
o.e
0.5
0.4 -
0.3 -
O.Z -
0.1 -
0 10
N-6 Alph«-3
HR. 5.8(b) Aulooorrolotlon Function va Timo
Amplitude
N-13 Alpha-2
fh-S.9(1t) Aulooorrelatlon Function va Time
1.2
1.1
1.0
0.0
O.S
0.7
o.e
0.5
OA
0.3
0.2
0.1
0.0
-0.1
-0.2
Amplitude
BZZ J I I I
yw. I ' I 1 1 I 1 1
i 8 10 12 14 16 18 20 22 24 20 26 30
TImo
N-13 AlpliB-3
Flg.5.J(c) Aulooorrelatlon Funollon V8 Time
Ampllluda 1.2
1.1
1.0
0.0
0.8
0.7
0.6
O.S
0.4
0.3
0.2
0.1
0.0
-0.1 h
-0.2
- -
A> 7\A /\
: :..yyy.. V -1 1 1 1 1 1
• . : : : : - • .
/ \ / V \ • N . / \
y.. yy\j : 1 1 1 1 1 1 1 1
e 8 10 12 14 ia 18 20 22 24 28 28 30
Tims
N-13 Alpha-2
n8.S.9(<l) Autooorrelation Funotlon va Time
Amplitude 1.2
1.1
1.0
0.9
0.8
o.r
0.8
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
y W' w , w v I I I ' I L l 1 I 1 J I |_
e 10 12 14 18 18 20 22 24 28 28 30 Tlma
N-13 AlphB-3
fig.S.9(n) Aulooorrelallon FunoKon va Time
Amplltudo
0.9
0.0
0.7
0.6
0.6
0.4
0.3
0.2
0.1
O
-0.1 h
-0.2
_
t
•
/\ A A A A A A A A A A A V \/ \/ V
1 1 1 1 1 I 1 1 1
—
—
10 IS 20 2 5
Ttma
30 35
N-21 •lpha>2
Flg.ioa Auloeorrelatlon Funotlon vs Time
40 45 5 0
AmpMluda 1
0.0
O.B
0.7
0.6
0.6
0.4
0.3
0.2
0.1
0
- 0 . 1
- 0 . 2 10 15 20 2 5
Time
30 35 40 45 60
N-21 alpha-3
Fig.10b Aulocorrolol lon F u n c t i o n va Time
Amplitude
•0 .2 10 15 20 2S
Time
30 36 40 60
N-21 alpha-2
FIg.lOe Aulocorrelallon Function v« Time
Amplitude
0.9
0.8 - -
0.7
0.8 -
O.S
0.4
0.3
0.2
0.1
0
-0 .1 -
•0 .2
y^z^ '•JwKIy J!^
10 16 20 25
Time
30 3 5 40 46 60
N-21 elpha-S
Flo-IOd Autocorro la l lon Func t ion v t Time
Ampllludo
0.9
0.8
0.7
• 0.8
O.S
0.4
0.3
0.2
0.1
0
- 0 . 1
- 0 . 2
.«ft A'^-V^/ Vy"\/ -v J - A v V ^ W ^ ^ -
20 40 80 80 100 Time
N-49 •lpha>2
Hy . 5.11(a) Autocorrelation Function va Time
Ampllluda
100
N-49 alpha-2
Pi|^ 5.11(b) Autocorrolntlon Function v« TImo
only as for binary or uniform sequence |a„| = 1 Therefore, the starting sequences in
the present work is taken as the sampled versions of the various window functions such as
Hann or Hamming window function and Kaiser window functions, [106]) (Kaiser
Window Function is selected for generating sequences because it realizes all other
window functions by varying its parameter a When a=0 Kaiser window function
corresponds to rectangular window function, a=5 444 it represents hamming window
function, and for a=8 885 it corresponds to Blackman Window Function) It is being
observed that the Kaiser window gives better results in comparison to other window
functions The three parameters for comparison are
(i) Energy Ratio or Energy Efficiency,
(ii) Side Lobe Energy,
(iii) Maximum Side Lobe Level
The Table 5 1 gives Huffman code along with its energy efficiency auto
correlation properties i e side lobe energy and max side lobe level The sequences
generated from Kaiser window function are given in Table 5 2, 5 3, 5 4 & 5 5 The
sequences in Table 5 2 (for a=2) and in Table 5 3 (for a=3) are optimized with sidelobe
energy Sequences in Table 5 4 (for a=2) and Table 5 5 (for a=3) are optimized with
optimization criterion which combined effect of max sidelobe level and side lobe energy
The different results of the above said tables have also been plotted in figures so as to
make the comparison among them easier
Figure 5 1 is the plot of energy ratio verses the sequence length Ihe energy ratio
of the sequences for a = 2 is always less than the energy ratio of the sequences for a=3,
if the same length of the sequences is considered But the trend of variation of the energy
ratio with the sequence length is same i e energy ratio decreases gradually with increase
in the sequence length upto a length of approximately 21 and then shows no appreciable
I deterioration
Figures 5 2, 5 3, 5 4 are plots of the side lobe energy and the max side lobe level
verses sequence length Figures 5 2 & 5 3 are for the sequences which are optimized
with side lobe energy and Figs 5 4 & 5 5 are for the sequences which are optimized
70
with both max side lobe level and the side lobe energy The parameter a=3 for figures
' 5 2 & 5 4, and a=2 for figs 5 3 and 5 5 When all the figures are compared, it is found
that the side lobe energy given in figures 5 2 & 5 3 is lower than the side lobe energy
given ni figures 5 4 & 5 5 at almost all sequence lengths The reason for lower side
lobe energy of figures 5 2 & 5 3 is that it is optimized for minimum side lobe energy
whereas the figures 5 4 & 5 5 are optimized with max side lobe level
The auto-correlation fianction has also been plotted for the different sequences of
lengths N = 5, N = 13, N = 21 and N = 49 Fig 5 8(b) & 5 8(c) represent sequences of
length N = 5, which are derived from Kaiser window for a=3 & a=2 respectively
The comparison of the sequences obtained with the above method and those
given by Huffman, shows that the results are not very good with respect to the side lobe
energy and the maximum side lobe This is so because of the fixed values of the
amplitudes of the sequence obtained by the window functions Search is confined to
changing the sign of the sequence element i e if sequence is given by ao, ai, a2, .an-i,
then in the search (using element complementation) the sequence is tested by changing
the sign of ao to -ao if the change decreases the functional then new value is retained
and the next element a2 is changed from a2 to -a2 and so on till the minimum is reached
A more general problem may be to search not only with respect to phase but magnitude
also, i e |a,| is also changed and the functional observed However, as the sequence
length N increases searching in both the directions phase and magnitude (in fact
magnitude for {a,} may assume very large number of values ideally infinite) becomes quite
cumbersome and one is not sure about the degree of improvements of the result In next
section however a simple approach for improving the energy efficiency of the multi-level
sequences is presented
5.4 CLIPPING METHOD
A figure of merit for the energy distribution of a sequence is the energy efficiency
defined by
7 = ^ T - T F ^ l (56) A^max„|fl„|
71
It is evident that the maximum value of r|will be attained for purely phase
modulated pulse trains such as binary sequences, having all element values equal to unity.
In this case the energy efficiency will be 100% as E will be equal to N (the length of the
sequence). The energy efficiency depends on two independent variables, namely, the total
energy E of the pulse tiain and the magnitude of the largest pulse denoted by inaxn|a(n)|
As can be seen iioin Eqn. (5.6) the energy efficiency of the Huffman sequences can
be increased by decreasing the energy contained in the largest pulse maxn|a(n)|. This is
done by process of clipping where the largest pulse is clipped to some level or in other
words to limit the total energy contained in the largest pulse. The clipping level is defined
is
_ m2Lx(Pulse.Height) - min(Pulse.Height)
100
It is the amount of clipping done in each step. The x is the percentage at which
the clipping is required, in the present case it is 5%. The clipped sequence so obtained
would not be a true Huffman sequence and its ACF would depart from being pulse like
Furthermore, the side lobe energy in the ACF of the clipped sequence will be greater
than that of the undipped sequence. As the clipping level is increased, the energy
efficiency of the resulting sequence increases, while* the side lobe energy also increases.
In the limit when the energy ratio is maximum, the resulting sequence will be a binary
sequence, but still having an ACF with small side lobe levels. The process of clipping
thus provides a means to generate sequences where energy ratio and side lobe energy
is a compromise between those of Huffman and binary sequences. What level of clipping
should be chosen would however, depend upon the actual requirements and test
conditions.
The process of clipping is applied to the HulTman sequences of length 13 and 32
and 128. These sequences were obtained using the method given in Section 5 275. The
values of these sequences are given in appendix. The results are given in Tables 5.6, 5.7
and 5.8 respectively. The lobes give the variation of energy efficiency and side lobe
energy for various levels of clipping. For the Huffman sequence of length 13, 32 & 128
these variations are also shown graphically in Fig 5 12a, b, c. It can be seen from the
lobes and figures that as the clipping level is increased the energy efficiency of the
resulting clipped sequence increases while the side lobe energy also increases For full
72
3: m 5 <D
(D — •
(D S-S-ZJ Oft -
S o ca
§•2.8-% ^ m • 2 . 2
_L " * (D
IS
o £
^ O f-
© e g
•a ^ ^ 55 o g
1 — 0 )
el"' a» k-c ©
•a
CM
c as
O E
S *- ^ c o T
LJJ C
CO O CD
el"' LJJ ::i
• • • •
•a
o o
o 09
O <0
••-• c
J> o CL
z * ^
o
Ob
s
Q
TABLE-5.6 : Energy-Ratio, Side-lobe Energy for Different levels of clipping of the Huffman sequence of Length 13.
S.No. Clipping level Energy Ratio Side-lobe Energy
8.46 0.020
9.10 0.0215
9.60 0.0248
10.09 0.0255
10.63 0.0275
11.07 0.0303
11.59 0.0350
11.97 0.0425
12.41 0.0540
12.71 0.0616
13.0 0.071
1 .
2 .
3 .
4 .
5 .
6 .
7 .
8 .
9 .
1 0 .
1 1 .
1 . 0 9 3
1 . 0 4 3
0 . 9 9 4
0 . 9 4 4
0 . 8 9 5
0 . 8 4 5
0 . 7 9 5
0 . 7 4 6
0 . 6 9 6
0 . 6 4 7
0 . 5 9 7
TABLE-5.7 Energy Ratio and Side-lobe Energy for Different Levels of Clipping of the Huffman Bequence of Length 32.
S.No. Clipping Level Energy-Ratio Side-lobe Energy
1 .
2 .
3 .
4 .
5 .
6 .
7 .
8 .
9 .
1 0 .
1 1 .
1 2 .
1 3 .
1 4 .
1 5 .
1 6 .
1 7 .
1 8 .
1 9 .
2 0 .
4 1 . 3 7
3 9 . 4 4
3 7 . 3 2
3 5 . 4
3 3 . 2 6
3 1 . 2 5
2 9 . 2 1
2 7 . 1 9
2 5 . 1 6
2 3 . 1 3
2 1 . 1 1
1 9 . 0 8
1 7 . 0 6
1 5 . 0 3
1 3 . 0
1 0 . 9 8
8 . 9 5
6 . 9 3
4 . 9 0
0 . 8 5
1 0 . 3 5
1 1 . 2 9
1 2 . 2 2
1 3 . 2 6
1 4 . 3 2
1 5 . 3 8
1 1 . 4 2
1 7 . 4 5
1 8 . 5 3
1 9 . 9 1
2 1 . 4 3
2 2 . 6 5
2 3 . 6 0
2 4 . 5 5
2 5 . 6 4
2 6 . 6 6
2 7 . 7 5
2 8 . 4 7
2 9 . 0 3
3 2 . 0 0
0.197 X 10
.-3
-4
0.54 X 10
0.28 X 10
0.70 X 10
-2
-2
0.129 X 10 -1
0,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0 ,
0 .
0 ,
. 02
. 0 3 2
. 0 4 4
. 0 5 8
. 0 7 9
. 1 0 6
. 1 3 1
. 1 5 3
. 1 8 6
. 2 2 8
. 2 5 8
. 2 9 0
. 3 1 2
, 3 3 8
. 4 6 1
clipping such that all the pulses have same value, the resulting sequences are the binary
1 sequences of corresponding length. It is interesting to mention here that full clipping of
the Huffinan sequence of length 13 results in the Barker sequence of same length. The
binary sequence of length 32 obtained by clipping the llufTman sequence of length 32 and
128 (Table 5.7 & 5.8) also has low side lobe energy.
Thus the method of clipping can be used to advantage for generating either
binary sequences or non-binary sequences having low side lobe energy and high energy
efficiency. The main requirement of this method is that the starting sequence which is a
non-binary sequence, must have very low side lobe energy. This method can, therefore,
be used only in those cases where non-binary sequences having good auto-correlation
properties are available or a method is available for the generation of such sequences.
5.5 ITERATIVE PROCESS FOR THE GENERATION OF MULTILEVEL SEQUENCES
5.5.1 Introduction
In this section the multilevel sequences are generated itteratively. It is observed
that the inverse of a sequence having low side-lobe energy has good ACF better than
' the sequence itself This fact is being utilized to generate the sequences of low side-lobe
energy by repeating the process of inversion a number of times. The inverse of a
sequence is obtained by obtaining the inverse filter of the same length The values of
the top gains of the inverse filter are taken as the new sequence which would have lower
side-lobe energy as compared to the first. Inverse filter coefficients are again obtained
using the new sequences to yield a sequence which would have still lower side-lobe
levels. Thus new sequences having lower side-lobe energy are generated iteratively by
successively taking the coefficients of the inverse filter as a sequence and computing the
corresponding inverse filter coefficients. In the following sections the problem of
inversion and the selection of the initial sequences, is considered in more details.
73
5.5.2 Optimum Inverse (Least Error Energy) Filter
The optimum mverse (or least error energy or spike) filter, [43] problem for
discrete time system can be formulated with reference to Fig 5 13 The error sequence
{e,} represents the dincrence between the desired output sequence {d,} and the actual
output sequence {c,} {a,} is the input sequence of length n+1 The desired output
sequence in the present case, consists of a unit 'spike' at some time index k The
optimization problem is to select the fiker weighting sequence {f,} of length m+1 such that
the error energy of the error sequence {e,} is minimized The error energy I is given by
m+rt m+n m
/=Zef=ZK-I/„a,_,r (5 7) t=o r=o «=o
The solution to this optimization problem is given by Robinson, [43], [77]
According to him, if there is no constraint on the admissible values of fj, j = 0, 1,2, N,
then the optimum filter coefficient must satisfy the relation 5I/c)fJ=0, for j = 0, 1, ,m
The result of performing the indicated operation is
T^Lrj-,=gj (5 8)
j=0, 1,2, ,m
where r _, = Sa,- ,« ,_^, is the auto-correlation function (for index, j-s) of the
2Ar-2
input sequence (a,), and gj = Y,d,a,_ , j=0, 1, ,m, is the cross-correlation of the 1=0
sequence {d,} with sequence {a,} since r+j = r-j V j>0, Eqn (5 8) provides a set of (m+1)
linearly independent expressions in (m+1) unknowns The minimum value of I, (obtained
by using fo, fi, fm, determined from the solutions of Eqn (5 8) is
i^,„-i.df-i:f,g, (5 9) t-0 1=0
The finite length optimum inverse filter for a given input sequence obtained
by solving the set of Eqn (5 8) is not an exact inverse An exact inverse would be of
infinite length, [77], [78] The filter, however, is optimum for the chosen length m+1 If
the desired length of the filter in Eqn (5 8) is increased it will approach the exact inverse
74
Input Sequence 1 Digital Filter
i-i)
Actual output
[fi) 1 >-
Desired Ovitput Sequence o
U,] ^ ^
Error Sequence
rig. 5J3
•o X
7 -
6-
5 -
*io
'-
1 -
U
)2
)0
! ^
u
c u 1
J3 3 -o
:* X)
8 o
• -
<« Q:
^ 6
(U c
UJ
E n e r g y Ra t io
S i d e l o b e E n e r g y
\ \
\
T 6
- T " 10 8 10 12
N u m b e r of c o m p u t e r r u n s
16
Fig 5 14 (d) Vanation of energy ratio and side-lobe energy at different iterations for tlie B.iikcr sequence of length 13
' o X
8-\ 11
10
64
c
X)
Ji 2H
(TJ
c Ul
— E n e r gy R a t i o
- S i d e - l o b e E n e r g y
0 0 /; 6 8 10
Number of compu te r r u n s
12 R 16
Fig. 5.14(b) Vanahon of energy ratio and side-lobe energy at different iterations for tlie Barker sequence of length 11.
' O
X
28-
30 E n e r g y Ra t io
Side- lobe Energy
2M 2M
20 20
>.
0)
c
i)
•D
'J)
16
TO
C7>
c
,12
8 -
i.
_y
T" 8 10 12
Number of computer runs
1^ ^1 ' 16
Fig. 5 14(c) Variation of energy ratio and side-lobe eiieipy ;it diffeient iteiutioiis for llie binary sequence of length 31
0.5-
O.i.
1.6-
hi4-
1-2-
10-'
8
7
6
5
0.3-
i_ a> c
r" 0;
^0.2-o -— •a, TJ
tn
0) c
LxJ
0) JO
O -J* o
•D
CO
O.A
0 . 1 -
0.2-
O p l i m u m p u l s e p o s i l i o n at 1
O p t i m u m . p u l s e p o s i t i o n at 15
O p t i m u m p u l s e p o s i t i o n at ^
(For B i n a r y S e q u e n c e o f l e n g t h 5)
T - , 1
c. 6 8 10
N u m b e r of c o m p u t e r r u n s
1 —
12 n— 1^ 16
fig 5 15 V.UK1IIOI1 of side-lobe energy at different iterations for the starting sequence chosen at random
TABLE-6.9 : Energy ratio and side-lobe energy at different iterations for Binary and Huffman Sequences of Different Lengths
Comp- Barker sequence Huffman sequence Binary sequence uter of length 13 of Length 13 of Length 31 Run
Energy Side-lobe Energy Side-lobe Energy Side-lobe Ratio energy Ratio energy Ratio energy
1. 13 0.071 8.50 0.02 31 0.28
2. 6.8 0.017 8.20 0.016 14.4 0.12
3. 7.87 0.0117 7.96 0.014 14.6 0.06
4. 7.18 0.0097 7.68 0.012 11.1 0.03
5. 7.36 0.0086 7.42 0.011 12.2 0.0029
6. 6.92 0.0079 7.22 0.010 9.8 0.023
7. 6.94 0.0072 7.02 0.0093 10.9 0.019
8. 6.66 0.0067 6.87 0.0085 8.98 0.016
9. 6.71 0.0062 6.71 0.0079 9.9 0.013
10. 6.50 0.0059 6.59 0.0073 8.41 0.012
11. 6.48 0.0055 6.46 0.0068 8.3 0.010
12. 6.26 0.0052 6.36 0.0064 8.02 0.009
13. 6.29 0.0049 6.29 0.0060 8.83 0.0085
14. 6.10 0.0046 6.17 0.0057 7.72 0.0075
15. 6.1 0.0044 6.08 0.0054 7.60 0.0070
TABLE-5.10 : Energy-Ratio and side-lobe Energy at Different Iterations for the Barker Sequences of Different Lengths.
Comp- Barker sequence Huffman sequence Binary sequence uter of length 5 of Length 7 of Length 11 Run
Energy Side-lobe Energy Side-lobe Energy Side-lobe Ratio energy Ratio energy Ratio energy
1 . 5 . 0 0 . 1 6 7 . 0 0 . 1 2 2 1 1 . 0 0 . 0 8 3
2 . 2 . 9 7 0 . 0 5 5 . 2 2 0 . 0 7 6 8 . 7 1 0 . 0 6 3
3 . 3 . 5 0 . 0 3 4 . 4 3 0 . 0 4 7 . 6 2 0 . 0 4 5
4 . 3 . 2 0 . 0 2 3 4 . 0 3 0 . 0 2 7 6 . 8 7 0 . 0 3 1
5 . 3 . 3 5 0 . 0 2 0 3 . 9 5 0 . 0 2 2 6 . 5 5 0 . 0 2 2
6. 3.32 0.016 3.93 0.0191 6.07 0.016
7. 3.69 0.013 3.96 0.0171 6.01 0.012
8. 3.38 0.012 3.99 0.015 5.72 0.0097
9. 3.42 0.010 4.02 0.013 5.73 0.0078
10. 3.45 0.009 4.06 6.012 5.53 0.0069
11. 3.33 0.008 4.10 0.011 5.56 0.0054
12. 3.20 0.007 4.14 0.0098 5.43 0.0046
13. 3.10 0.0067 4.18 0.0091 5.47 0.0039
14. 3.00 0.0061 4.22 0.0084 5.35 0.0035
15. 2.92 0.0Q56 4.26 0.0077 5.40 0.0031
TABLE-5.11 : Energy-Ratio and Side-Lobe Energy of the Sequence of Length 20, Generated from the Barker Sequence of Length 13.
Computer Run Energy-Ratio Side-Lobe Energy
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
13 (Barker sequence 0.071 of Length 13)
Sequence of Length 20
5 . 7 1
7 . 1 8
7 . 8 7
7 . 1 7
7 . 5 6
7 . 3 7
7 . 4 6
7 . 6 1
7 . 4 1
0 . 2 6 8
0 . 4 1
0 . 1 2 6
0 . 3 2
0 . 0 9 5
0 . 2 7
0 . 0 8 1
0 . 2 3
0 . 0 7 2
TABLE-5.13- Energy-Ratio and Side-Lobe Energy of Sequences of Length 13 and 5 obtained by Choosing the initial sequence at Random (Optimvim Pulse Position not at nT second)
Computer Run
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Optimum tion at All the
(a) Pulse posi-T sequence-elements
having values +1
Energy Ratio
13
2.01
5.03
1.85
• 4.65
1.85
4.51
1.85
4.44
1.85
4.39
1.85
4.36
1.85
4.33
Side-Lobe energy
7.69
0.50
5.71
0.50
5.2
0.5
4.97
0.50
4.86
0.50
4.79
0.50
4.24
0.50
4.70
Optimum (b)
1 Pulse position at 15T-sequenc
1, 1, --1, -1,
Energy Ratio
13
3.15
6.3
3.13
5.59
3.1
5.4
3.2
5.1
3.12
4.89
3.07
4.7
3.04
4.61
e 1,-1, 1, 1, 1, 1, 1, 1, -1.
Side-Lobe energy
1.58
0.80
0.82
0.71
0.68
.69
0.60
0.67
0.55
0.66
0.52
0.64
0.49
0.63
0.47
(c) Sequence of Length 5, having optimum Pulse position at 4T
Energy Ratio
5.0
1.77
1.96
2.72
2.02
2.02
2.06
2.58
2.1
Side-Lobe energy
0.479
0.33
0.33
0.372
0.36
0.36
0.38
0.42
0.396
and, therefore, the error energy in Eqn (5 7) and the minimum mean square error in
Eqn (5 9) would decrease Robinson, [43], [77] has also mentioned this He further
comments that, for a given sequence the least error energy depends very critically
upon the lag of the spike in the desired output, relative to the input The optimum lag
depends upon the delay properties (maximum, minimum, or mixed delay) of the input
sequence
If the length of the spike filter (m+1), is made equal to the length of input sequence
(n+1), then the (n+1) equations (Eqn 5 8) can be written in matrix form as
RA = G (5 10)
Where A is a column rector containing (n+1) weighting sequences of the filter, g is a
vector containing (n+1) discrete cross correlation function of the input sequence (a,}
and the desired output {d,} R is the (n+1) x (n+1) auto-correlation matrix, each row of
which consists of a shifted version of the auto-corrclation of the input sequence (a,) and is
given by
R = 'n-l (5 11)
If the input sequence has near a ideal auto-correlation function (such as the
Huffinan sequence), then r, = 0 for j ?t 0 or n, and also ro » rn, therefore, neglecting rn,
the auto-correlation matrix R becomes a diagonal matrix and the filter coefficients fj from
Eqn (5 10) aic given by
fj = gAo,j = 0, 1, ,n (5 12)
If the spike in the desired output sequence lies at time index (n+1), the cross-
correlation coefficients are
gj = a„.j,j = 0, 1,2, (5 13)
75
FromEqn (5 12) the spike filter coefficients are
fi = a„.j/ro,j = 0, 1, ,n (5 14)
Which shows that in this special case the spike filter is the filter matched to the
input sequence {a,} Hence, with sequences having near ideal ACF The spike filter is
almost identical to a matched filter Eqn (5 12) fijrther suggests that to make the filter
energy small, i e better performance in the presence of noise, ro (the total energy of the
sequence) should be large This can be done by choosing the sequence of greater length
Robinson, [43], [77] gives a most usefiil set of Fortran programmes for efficiently
solving such matrix equation and computing the coefficients of the least error energy
filter for optimum spike position The listing of the programme is given in Appendix
5.5.3 Results and Discussion
The side lobe energy of the sequence {a,}
(SLE)^=2.tr,' (5 15)
If fj is taken as the input sequence (eqn 5 14) then the side lobe energy of the
sequence { }
(^L£)^=2.I r ;^ = 2 . I ^ (5 16) . 1 . 1 "
Where r', are the side lobes of the auto-correlation function of the sequence {f,)
for n > 1 , ro > 1 , therefore
(SLE)F < (SLE)A (5 17)
This condition is derived when rj = 0, for j ; ! ! : 0 or n In practice rj O for j T 0 or
n, i e the auto-correlation function always has some side-lobes, however, if the main
lobe IS of coiisidciably greater magnitude than the side-lobes, the condition derived in
76
Equ. (5.17) will still hold good, though the filter coefficients ^ will not be exactly equal
to the value given by Equ. (5.14), [98].
From the above discussion it can be concluded that the inverse of a sequence
having low sidelobe energy should have a good auto-correlation function, better than the
sequence itself. This fact is being utilized to generate a sequence of low side-lobe energy
by repeating the process of inversion a number of times. The inverse of a sequence is
obtained obtaining the inverse filter of the same length. The values of the tap gains of the
inverse filter are taken as the new sequences which would have lower side-lobe energy as
compared to the first. Inverse filter coefficients are again obtained using the new sequences
to yield a sequence which would have still lower side-lobe levels. Thus new sequences
having lower side-lobe energy are generated iteratively by successively taking the
coefficients of the inverse filter as a sequence and computing the corresponding inverse
filter coefficients The initial or the starting sequence for this iterative process can by any
sequence of desired length, having low side-lobe energy.
As an example of the above process, the Barker sequence of length 13 is taken as
the starting sequence. The side-lobe energy of this sequence is 0.071 whereas the side-lobe
energy of the corresponding inverse filter coefficients is 0.017 which is much smaller than
the side-lobe energy of the original Barker sequence. The sequences are here normalized
to have unit energy Table 5.9 gives the energy ratio and the side-lobe energy for the
Barker and Huffinan sequences of length 13 and the binary sequence of length 31, at
different iterations or computer runs It can be seen from the table that indeed the side-lobe
energy in all three cases decreases witii the number of iterations or computer runs
undesirable elVect is that the encigy ratio of the lesulting sequences also decreases
However, the decrease in energy ratio is not much as compared to the decrease in the side-
lobe energy. For example, in the case of Barker sequence of length 13, and for 15th
computer run the value of the side-lobe energy decreases fi"om 0 071 to 0 004 while the
energy ratio reduces from 13 to 6.12. The side lobe energy of the 31 length binary
sequences is reduced from 0.28 to 0.007 Hence the sequences having very low side-lobe
energy can be generated using this method
In all the cases shown in Table 5.9 & 5.10, the side-lobe energy is decreasing with
the number of iterations. In other words the iterations process is said to be converging. In
liie picsciil woik the word 'coiiveige' would be used in this scuijc and would mean that the
side-lobe energy decreases with the increase in the number of iterations.
For the Barker sequence of length 13 and 11 and for the binary sequence of
length 31, the variation in the side-lobe energy and the energy ratio with the number of
computer runs (iteration) is shown in Figs. 5.14 a, b & c respectively. It can be seen
from these figures that the iterative process converges in each case. Both the side-lobe
energy and energy ratio tend to become constant after about 15* iteration. The
maximum improvement in the side-lobe energy takes place during the first few iterations
after which there is only a slight improvement.
In all the above cases the length m of the optimum inverse filter was kept equal
to the length of the initial sequence n, thus the new sequence generated has the same
length as that of the starting sequence. If the length of the optimum inverse filter is
increased so that it is greater than the length of the starting sequence, a sequence of length
greater than the original sequence and equal to the length of the filter will be generated.
This new sequence can now be taken as the starting sequences for iterative process of
section 6.2. This would thus, provide a method to generate sequences of lengths other
than that of the chosen starting sequence. As an example, coefficients of 20 length
optimum I.F., were computed for the Barker sequence of length 13. These 20
coefficients of the fiher formed the elements of the new starting sequence of length so.
The resuUs of the iterative process are shown in Table 5.11. It can be seen from the table
that the resulting sequences of length so, do not have 100 side-lobes levels. Furthermore,
the convergence rate of the process is very small. This suggests that in order to obtain
sequences having good correlation properties the length of the filter should be the same
as the length of the starting sequence.
78
5.6 CHOICE OF STARTING SEQUENCE
The present section is devoted to study the consideration necessary in choosing
the starting sequence for the iterative process. In other words to find out necessary
condition to be fulfilled by a sequence so that it can be taken as the initial sequence
In section 6 2 and 6 3 Barker sequence of different length and binary sequence
of length 31 were taken as the initial sequences. The iterative process converged in all
these cases. These sequences possess good ACF properties in that their side-lobe energy
is low It would not be out of place here to check, if convergence is achieved with any
sequence chosen at random. Also if this does not works, what other constraints must
be imposed on the starting sequence so that the iterative process converges
5.6.1 Choice at Random
A sequence of length 13 having all elements values equal to +1 was taken as
starting sequence for the iterative process The results are given in Table 5 12 It is
observed from the table that the process does not converge but oscillates between two
values. Similarly a few other sequences were chosen at random The results are also
given in Table 5 12 Again convergence is not obtained, in these cases The variation of
side-lobe energy will the number of computer runs for these sequences are shown in Fig
5.15. It can be seen from the figure that the process does not converge in any of the
cases, and even diverges in some cases
5.6.2 EfTect of Optimum Pulse Position
Comparison of results obtained with Barker sequence and these given in Table
5 13 where the choice of the starting sequence was made at random, show that
convergence is achieved in all these cases where the optimum pulse position in the
response of the I F , to the sequence lies at nT second It, therefore, appears that
sequences for which the optimum pulse position in the output of the inverse filter occurs
at nT seconds, should provides better results This argument seems to be more logical
since the main lobe in the ACF of a sequence occurs at a delay of nT seconds where n is
the length of the sequence and 2n-l is the length of the ACF of the sequence Thus the
79
initial sequence should be such that the optimum pulse position in the output of the
inverse filter should coincide with the main lobe in the ACF of the initial sequence.
For the initial sequences chosen at random in section 6.5.1 the optimum pulse
does not occur at the instant nT in the output of I.F. The optimum pulse position in these
cases occurs at T and 15 T. In both these cases convergence is not proper.
For Barker Sequence taken as initial sequences for the iterative process, the
optimum pulse position in the ' output of IF., occurs at nT. For Binary sequence of
Length 3 land Huffman sequence of length 13, the optimum pulse also occurs at the
instant nT. The iterative process converges in all these cases. It, therefore, appears that
the sequence for which the optimum pulse position in the output of the IF. occurs at
the instant nT, do converge and other sequences do not converge. It is possible
while designing the optimum inverse filter, to deliberately fix the position of the pulse
in the output, at the instant nT. For example, in designing the IF . for the sequence of
length 13 having all values+1, the pulse position in the output of the IF is fixed at 13
T. The results of the operation are shown in Table-5.12. It is observed fi^om the table that
the results obtained are not satisfactory. The side-lobe energy reduces to zero, while
the energy ratio reduces to one, i.e. a single pulse of unit value is left at the input.
In the light of the above discussion it appears that a good initial sequence for the
iterative process is the one for which the optimum pulse position in the response to the
corresponding IF. naturally lies at the time instant nT, where n is the length of the
sequence.
5.6.3 Conditions for Convergence
From the discussion in the preceding sections, it is clear that for the iterative
process to converge, the starting sequence should be such that the optimum pulse position
in the response of the optimum IF. should occur at the time instant nT. It may be
because of the fact that the main lobe in the ACF of sequence occurs at the instant nT.
Furthermore, the weighting sequences of digital matched and optimum IF are almost
identical when the ACF of the sequence is near ideal. It is also necessary for
convergence to be achieved that the length of the optimum I.F. be made equal to the
80
length of the sequence. As has been observed in the results presented in the previous
sections, the iterative process converge in all such cases when the starting sequence
fulfills the above mentioned conditions
5.7 AVERAGING THE SPIKING FILTER COEFFICIENT WITH THE INPUT CODE
In this section non-uniform or multi-level sequences having pulse like auto
correlation function are generated by applying the linearity property of the Fourier
transform to a code and the inverse of the code in order to average the spectrum The
inverse of the code is obtained by inverse or least square energy filter (section 5.4) The
following well known properties of the Fourier Transform are used
A) Linearity Property (Superposition)
if g,(t) <^ G,(f) & g2(t) o G2(f)
then
a g,(t) + b g2(t) o a G,(0 + b G2(f)
where a & b are constants
B) The energy density spectrum and auto-correlation function form the F T pairs
Rg(t) o IG (f)|'
C) F [0(1)] =1 or constant i e 8(1) o i
As can be seen from the relation B, the auto-correlation function and energy
density spectrum form the Fourier Transform pair For an ideal auto-correlation
function, it should have a narrow peak at the centre and zero elsewhere as shown in the
Fig 5 16. With reference to relation C, the Fourier Transform or spectrum of this type
of function is a constant i e flat or uniform To obtain such type of auto-correlation
function the code should also be a impulse However, this is not possible, the
transmitted energy can not be concentrated in a single pulse, it has to be distributed over
81
some interval of time. Therefore, the spectrum of the actual transmitted signal departs
from being a uniform or flat one. If the spectrum of the input code {a;} is A(f), the
spectmrn of its inverse code {f,) will approximate 1/A(f) in the least square sense The
resultant or average spectrum of these two codes will be :
Uif) + h ^ (5.18)
This spectrum will be closer to being flat or uniform spectrum in comparison to
the spectrum A(f) of the input code {a;}. Hence using the linearity property of the FT.
from the relation A, following result is obtained :
l{ai} + l{fi}<^hA{f) + j±jy (519)
The average code {b;} given by i{«,} + ! { / } will have spectrum which will be
closer to being a uniform spectrum in the least square sense. Since the energy density
spectrum and the auto-correlation flmction form the Fourier Transform pair, therefore,
the average code {b;} corresponding to the LHS of the eqn. (5.19) will have better auto
correlation flmction than the input code {ai}. This can also be shown with reference to
Eqn. (5.14) by the analysis given below. The average code {bi} is given by the following
equation
The side-lobe energy of the code {aj} is
iSLE),=2.ir^ i = l
and the side-lobe energy of the average code {bi} is
(SLE)„ = 2.±r:' 1=1
where r'iare the auto-correlation side lobes of the code {bi}
82
TABLE-5.14
Initial Code {a.} Corresponding Average Code {b.}
1 ) Binary Code of Length 5 All elements ( +1 ) Energy Ratio = 5 SLE =2.39 Max SL =0.8
Energy Ratio = 2.5 SLE =1.20 Max SL =0.43
2) Barker Code of Length 5 Energy Ratio = 5 SLE =0.16 Max SL =0.2
Energy Ratio =3.72 SLE =0.05 Max SL =0.15
3) Barker Code of Length 7 Energy Ratio = 7 SLE = 0.122 Max SL = 0.143
Energy Ratio = 6.0 SLE =0.09 Max SL = 0.137
4) Barker Code of Length 11 Energy Ratio = 11 SLE = 0.082 Max SL =0.09
Energy Ratio = 9.74 SLE = 0.071 Max SL =0.08
5) Barker Code of Length 13 Energy Ratio = 13 SLE = 0.071 Max SL = 0.077
Energy Ratio = 9.08 SLE = 0.017 Max SL = 0.063
6) Huffman Code of Length 13 Energy Ratio = 8.5 SLE =0.02 Max SL = 0.099
Energy Ratio =8.32 SLE = 0.017 Max SL = 0.091
7) Huffman Code of Length 32 Energy Ratio = 10.35 SLE = 0.23 X 10 Max SL = 0.099
-4 Energy Ratio = 10.35 SLE = 0.19 X 10 Max SL = 0.091
-5
8) Binary Code of Length 31 Energy Ratio = 31 SLE =0.28 Max SL = 0.129
Energy Ratio = 20.0 SLE =0.13 Max SL = 0.095
9) Binary Code of Length 31 Energy Ratio = 31 SLE =0.39 Max SL = 0.193
Energy Ratio = 13.87 SLE =0.18 Max SL =0.185
10) Binary Code of Length 21 Energy Ratio = 21 SLE = 0.245 Max SL = 0.143
Energy Ratio = 11.2 SLE = 0.107 Max SL =0.113
11) Binary Code of Length 31 Energy Ratio = 31 Energy Ratio = 17.64 SLE = 0.14 SLE = 0.052 bjlax SL = 0.096 Max SL = 0.067
12) Binary Code of Length 41 Energy Ratio = 41 Energy Ratio = 26.65 SLE =0.14 SLE = 0.052 Max SL = 0.097 Max SL = 0.065
13) Binary Code of Length 53 Energy Ratio = 53 Energy Ratio = 31.32 SLE = 0.218 SLE = 0.119 Max SL = 0.094 Max SL = 0.091
TABLE-5.15
Barker code Average code {b.} of length 13 derived from Barker of length 13 derived from Huff
Huffman code Average code {b.}
code of length 13
Amp. Ph. Amp. Ph. Excit. Excit. Excit. Excit.
(Degree) (Degree)
man code of lengt
Amp. Ph. Amp. Ph. Excit. Excit. Excit. Excit.
(Degree) (Degree)
0
180
180
0
180
180
0
0.90
1.05
0.90
1.05
0.90
1.05
1.19
1.05
0.90
1.95
0.90
1.05
0.90
180
180
0
180
180
1.0 0
1.093
0.597 0
0.791
0.686
0.905 180
1.086 180
0.905
0.686
0
0.791 180
0.597
1.093 180
1.0
0.96
1.10
0.60
0.80
0.69
0.92
1.10
0.92
0.69
0.80
0.60
1 . 10
0.96
0
0
0
0
0
180
180
180
0
•RCt)
t (0^)
^CO
( t )
Pig. 5.16
-4
11
I—«
09
- • [
to
>
21 09'
>
iSLE),=[^]\2.Tr^ i= l
(SLE),=[^\\(SLE)^
f o r / i > l ; i ^ < l
Therefore
(SLE)B < (SLE)A (5.20)
This condition is derived when TJ = 0 fytj^O or n. In practice rj O for j ^ 0 or n i.e.
auto-correlation function always has some side lobes, however, if the main lobe is of
considerably greater magnitude than the side lobes, the condition derived in Eqn. (5.20)
will still hold good, though the filter coefficients, {f;} will not be exactly equal to the value
given by Eqn. (5.14).
Table 5.14 gives the energy ratio, side-lobe energy and the maximum side-lobe
for the various codes of different lengths. Table 5.15 gives the excitation values for some
of the initial codes and the codes derived from them using proposed method. The Code
No. 6 and 7 in Table 5.14 are the Huffman codes of length 13 and 32 given by Ackroyd,
[75], [79]. Code No. 8 is a binary code of length 31 given by Chakraborty, [63]. Code
No. 9 is a maximal length shift register sequence. Whereas the codes from No. 10 to No.
13 are taken from Chapter 4. As observed from the table the side-lobe, energy and the
value of maximum side-lobe of the derived average code (b;) is smaller than the codes
from which they are derived i.e. code {a;}. This improvement is achieved in all the cases
presented in Table I. In some cases there is significant improvement. For example, the
binary codes of length 31 and 41 has side-lobe energy 0.14 and 0.133 respectively,
whereas the derived average code {bi} has side-lobe energy 0.052 in both the cases.
Similarly the maximum side-lobe of these initial codes are 0.096 and 0.097, whereas the
derived average codes have maximum side-lobe equal to 0.067 and 0.068 respectively.
The improvement in auto-correlation fijnction can also be seen with reference to Fig.
5.17, which depicts the auto-correlation fianction of binary code of length 31 and the
auto-correlation fijnction of the corresponding code {bi}. The value of the auto
correlation ftinction at each time shift is smaller in the case of code {bi}. The same is the
83
case of Barker code of length 13 as the initial code whose auto-correlation function is
given in Fig 5 19 The corresponding average code {b,} has smaller auto-correlation
side-lobes as can be seen in Fig 5 20 Fig 5 21 shows the auto-correlation function of
Huffinan code of length 13 The maximum side-lobe of this code is 0 099 whereas the
corresponding code {b,} has the maximum side-lobe equal to 0 091
From the above discussion it can be concluded that the basic idea of averaging the
input code with the coefficients of the spiking fiher provides a useful method of
generating the non-uniform codes of any desired length with good auto-correlation
properties
5.8 SUMMARY
In some circumstances the use of pulse trains, other than purely phase modulated
ones, may be precluded due to the expense incurred in providing A M However, the
additional expense of encoding and decoding in amplitude and phase may be justified for
radars that must cope with land clutter or operate in a dense-target environment
The very low side-lobes that can be achieved with am ph m pulse trains
presented in this chapter make their use particularly attractive in systems requiring a
large dynamic range Moreover, the excellent self-clutter rejection performance is
obtained without sacrificing close target seperability (no main lobe widening) This
property is, however, achieved at the expense of a significant reduction in power
utilization at the transmitter i e low energy efficiency of these pulse trains The loss in
energy efficiency of these codes or pulse trains is mainly attributed to the high degree of
side-lobe suppression In some applications a complete suppression of the side-lobes
may not be required, instead they can be kept to a specified low level Consequently,
the clipping method or averaging method of spiking filter coefficients with the input
sequence may be used to advantage for improving the energy efficiency
84
CHAPTEE-6
SIDELOBE REDUCTION
85
6.1 INTRODUCTION
In a dense target environment or in situations where there are large undesired
scatterers (point clutter), it is often desirable to reduce the time sidelobes of phase
coded sequences to a prescribed low level. In principle, there is no difference between
the problems of resolving a target in the interference from other targets and the
detection of target in clutter. In previous chapters the reduction of the range sidelobes
of the compressed pulse has been of much concern in the application of matched filter
techniques to radar systems. In fact the mutual interference between targets or self
clutter imposes rather fundamental limitations on resolution performance. So far, the
reduction of the range sidelobes has been approached via waveform design. However, the
attainable sidelobes levels for phase coded pulse trains might be inadequate for specific
applications. Although it isi possible to use a.m.ph.m. pulse trains such as Hufiman
Codes (or codes discussed in previous chapter) to obtain the desired degree of
discrimination, in most high power radar systems this method is not readily available.
Thus, the designer has to resort to other sidelobe reduction techniques.
In principle sidelobe reduction can be achieved by either
(i) Amplitude-or Phase weighting in the Frequency Domain
(ii) Amplitude-or Phase weighting in the Time Domain
The weighting may be accomplished at the transmitter or receiver or at both.
Furthermore, the shaping can be performed at the RF, IF or video stages.
Sidelobe suppression in the frequency domain requires the design of a filter
such that the spectrum of the filtered waveform has a hnear phase-frequency
relationship and that the spectrum magnitude be proportional to one of the many
available weighting functions such as Taylor or Chebyshev, [22], [31]. The rapid advance
of digital hardware, alongwith the pipeline FFT configuration, does permit practical
realization of the required transfer function as depicted in Fig. 2.9.
Amplitude weighting may be introduced into a pulse-compression system
either entirely at the receiver, entirely at the transmitter, or at both simultaneously. Equal
weighting at both the transmitter and receiver is equivalent to altering the transmitted
86
waveform. In this case the system is still considered as matched. However, it can be
shown that in the peak power limited case, the SNR which assumes weighting at the
receiver alone is greater or equal to the SNR which assumes matched weighting at the
transmitter and receiver, [35]. An additional reason for unilateral weighting at the receiver
is due to the advantage of operating the transmitter at its peak power limit (no
expensive amplitude modulators required). Furthermore, amplitude weighting solely at the
receiver can be maintained conveniently, due to the accessibility of the components and
the low power levels involved. For these reasons it is henceforth assumed that weighting
is performed solely at the receiver at the expense of a lower SNR.
A convenient way of weighting is at the IF stage in the time domain. Most of the
sidelobe reduction techniques which have been proposed depend on cascading a
weighting fiher (tapped delay line) after the MF or by providing a suitable band
shaping network, [76], [96]. However, instead of placing a sidelobe reduction filter after
the MF it is probably more straight forward to design a mismatched filter (MMF)
under some conditions of optimality, [78]. The amount of mismatch from the matched
conditions is usually characterized by the loss factor Ls, given by, [76]
SNRjweighted) ^' ~ SNR{matched)
For an input sequence a(n) of length (N+1) and a filter weighting sequence h(n) of
' length (M+1), Ls becomes
2
L. =
M
Y.Kn).a(j - n) n=0
I |a(ii) | Z|/i(/i)| «=0 11=0
^ 1 (6.1)
' Where j^N denotes the time delay for which the output is maximum. (It is assumed
that (M+1)>(N+1). The basis of Eqn.6.1 is that for coherent summation signal
I components add as voltage levels while the noise components add as power levels.
87
6.2 INVERSE FILTERS
The reduction of the sidelobe interference can be accomplished through the
use of inverse or deconvolution fihers. These filters (equalizers) have been of much
concern in removing inter symbol interference in communication and they are also of
interest in spectrum analysis. What is required ideally is a digital fiher which is inverse
to the code sequence. That is, the response of the filter to the code would ideally be a
sequence of which all the elements except one are zero. The position, L, of this nonzero
element, which can be chosen to have unity magnitude, is immaterial. If the code
sequence is (ao, ai,... .aw) its Z-transform is given by
A(Z) = ao + a,Z-'+ +aMZ-^
The filter will have the desired response if the transfer functions Z'VA(Z). If
A(Z) has all its zeroes within the unit circle, Z"VA(Z) will have all its poles there and the
filter will be stable and the transfer function will be realizable by a recursive filter or it
can be closely approximated by a transversal filter. However, it has been shown in the
preceding chapter that sequences with high energy efficiency (such as binary
i sequences) must have approximately half of their zeroes inside and half outside the unit
circle. The inverse filter will thus have poles outside the unit circle and consequently will
be unstable. Thus Z'^IA.{Z) will not be physically realizable for the signals which are of
interest as it will have poles outside the unit circle. Nevertheless, provided that L is large
enough, a good approximation to Z'VA(Z) can" be obtained which is physically
' realizable.
One approach is to factor 1/A(Z) into the product of a non-physically realizable
part 1/A.(Z), whose poles are all outside the unit circle, and a realizable part 1/A+(Z)
whose poles are all within the unit circle. This factorization can be done by the use of a
I polynomial root finding routine and a digital computer. The task, however, becomes
onerous for sequences of length more then about forty. 1/A-(Z) can be expanded by
polynomial division into a convergent series in powers of Z and if the expansion is
truncated afler the term in Z'', then the result, after multiplication by Z'^, is a
physically realizable transfer function. 1/A+(Z), having all its poles within the unit circle,
88
can be directly realized as the transfer function of a recursive filter. Alternatively, it can
be approximately realized by expanding it as a convergent series in powers of Z"
which is truncated at some point and then approximately realized as the transfer
function of a non-recursive filter.
Another approach to obtaining, to a degree of approximation, the weighting
sequence of an ideal inverse filter is to augment the sequence (ao, ai, , aM) to (P+1)
points with a large number of zeroes and to compute the discrete Fourier transform
(Ao, Ai, Ap) of the augmented sequence. The sequence (1/Ao, 1/Ai, 1/Ap)
represents the sample values of the transfer function of the inverse filter. By computing its
inverse discrete Fourier transform an aliased or folded version of the ideal inverse filter
weighting sequence is obtained.
A filter designed in one of these ways has a weighting sequence which is either
a truncated or a folded version of the weighting sequence of the ideal non-physically
realizable inverse filter. However, for a given weighting sequence length it is possible
to design filters with a lower mean square sidelobe level than can be achieved by this
method.
Robinson and Trietel, [77] have comprehensively studied the problem of
designing a non-recursive digital fiher so that its response to a given input sequence
approximates a desired output sequence in the least squares sense. In the present case
the ideal output is a sequence whose elements are all zero except for one unity element
occurring at some position L. The actual response of a fiher having the weighting
sequence (fo, fi, ...., fw) to the input (ao, ai, , SLM) will be some sequence (Co, Ci, ,
CM+N)- What is required is that the sum
V = C^+C^+ + ( Q - i ) ^ + C M+N
should be minimized by proper choice of the weighting sequence. The sum V
represents the 'energy' of the difference between the actual response and the ideal
response sequences. Robinson and Treitel show that V is minimized when the weighting
sequence is given by the solution of the vector matrix equation.
Rf = g
89
Where f is a column vector whose elements are the elements of the weighting sequence
'that is to be calculated and g is a vector containing the discrete cross-correlation
function of the input sequence and the desired output sequence. R is a matrix each row of
which consists of a shifted version of the auto-correlation of the input sequence (Section
5.5.2).
Robinson, [43], [62] gives a most useful set of Fortran programs for efiRciently
solving such matrix equations. In addition he gives a routine which computes the
optimum filter and evaluates V for each position L of the response peak. His programs
thus enable one to find the response peak position L for which the least squares inverse
filter of weighting sequence length M produces the best approximation to the desired
response and to compute the filter weighting sequence.
There are three characteristics of a sidelobe suppressing pulse compression filter
which are of particular interest. One is the total response sidelobe energy given by
M+N
k=0
Where filter weighting sequence has been normalized so that the peak of its
response to input sequence is unity. In a dense uncorrected target environment, the self
clutter power at the filter output is proportional to this quality. Another property of
concern is the peak sidelobe level, max |Ck|, which determine the ability of the system to
resolve a weak target which is adjacent to a strong one. Finally, the total 'filter
energy', given by
N
v. = I// *=0
is important since with white noise at the receiver input the noise power at the filter
output is proportional to this quantity.
As the optimum filter weighting sequence length is increased, its behaviour
approaches that of the ideal inverse filter (i.e. response sidelobes are suppressed), [78].
The Table 6.1 shows the decrease in signal to noise ratio (i.e. loss factor L„ equ 6.1)
That results when the matched filter is replaced by the ideal inverse filter for the Barker
90
sequences of Length from four to thirteen. It is to be remarked that a loss of only 0.2dB in
SNR is incurred when the sidelobes of the 13-element Barker sequence are completely
suppressed a fact which was first pointed out by Key et al, [76] and Messe et al, [96].
6.2.1 An Iterative Method for Range Side Lobe Suppression for Binary Codes
In section 5.5.2, it has been shown that the inverse filter weighting sequence or
coefficients obtained in the manner as discussed above, has bitter autocorrelation
flinction, smaller than that of the input sequence itself, [98], [99]. The same concept is
used here for sidelobe suppression by taking the filter coefficients as the input
sequence, evaluating its inverse filter and each time evaluating its response.
Table 6.2 and Fig. 6.1, show the variations of three quantities of a sidelobe
suppressing filter (V„ max C(k) and V„) for each iteration or computer run, when the
input sequence is the Barker sequence of length 13. The corresponding matched filter
output is also shown for comparison. It can be seen fi-om the table and the fig. that
quantities V, and max|C(k)| are decreasing with the number of iterations. The quantity
Vn is approximately constant. It can also be observed that as the iteration is increased
the performance of inverse filter and the matched filter becomes almost identical. The
advantage of the method proposed for sidelobe suppression is that the desired
suppression is achieved without increasing the length of the filter. The proposed
scheme can also be used for codes other than Barker codes.
Tabic; 6.1
Barker Sequence Length ^ ^ '" SNRwhen Matched Filter is
replaced by Inverse filter (dB)
4 1.7
5 0.6
7 1.5
11 1.5
13 0.2
91
24
to'6 X
UJ -J
in
r 8 S
0
70
- 60 tn
9 50 H
i/iO K
- J
-;^20 o E
10
26
20
7 o t 6 K
6 8 10 12 14 iloration number
16 _ i I -18 20
Ki);.E4 >'iiriofi()/i of I',, K, and mov | C , | <i( dijjerent iteraUuns Jor Barker atile oj leiuilh li
• sidclobe energy III'" oiilpiil): y, • sidclobc ciicrpy (corresponding Ml") X mnx sidclobc (II'): max | C, | • liller energy (IP); V,
Iteration
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
IF output SL energy V,
0.026
0.010
0.00858
0.0075
0.00717
0.0067
0.0062
0.0059
0.0055
0.0052
0.0050
0.0047
0.0045
0.0043
0.0041
0.0039
0.0038
0.0036
0.0035
0.0034
TABLE
max SL (IF)
max 1 Cj^ 1
0.063
0.055
0.056
0.052
0.052
0.049
0.048
0.047
0.046
0.045
0.044
0.043
0.042
0.041
0.040
0.039
0.038
0.038
0.037
0.036
6.2
Matched filter
0.071
.0.017
0.0117
0.0097
0.0086
0.0079
0.0072
0.0067
0.0062
0.0059
0.0055
0.0052
0.0049
0.0046
0.0044
0.0042
0.0040
0.0038
0.0037
0.0035
max SL (MF)
0.077
0.063
0.061
0.052
0.053
0.050
0.049
0.048
0.046
0.045
0.044
0.043
0.042
0.041
0.040
0.040
0.039
0.038
0.037
0.037
Filter energy V„ (IF)
0.980
0.985
0.985
0.986
0.987
0.987
0.988
0.988
0.989
0.989
0.990
0.990
0.991
0.991
0.991
0.992
0.992
0.992
0.993
0.993
6.3 SIMPLE SCHEME FOR SIDE LOBE ELIMINATION
In this section, a simple approach is developed which completely eliminates the
effects of the sidelobes.
Let {ai} is the input sequence to a matched fiher, matched to the input sequence
{ai} with discrete weighting sequence {fi}. With reference to Fig. 6.2, the output of the
matched filter {Ci} is the convolution of {a;} with {f;}. In terms of Z-transform.
A(Z) = ao + ai Z-' + aw-i Z"^"' (6.2)
F(Z) = aN-i + aN-2Z- + ... + ao Z"* "' (6.3)
C(Z) =A(Z)F(Z) =Co +CiZ- +C2Z- ... +CN.IZ-^^'+ +C2N-2 Z' " (6.4)
C(Z) is also the Z-transform of the auto-correlation function of the sequence {aj}
which is of the length (2N-1). The main lobe in the output of the filter occurs at a delay of
(N-1) T seconds or in the Centre, 1/T being the clock frequency. The output should
ideally be a pulse at t = NT seconds and zero elsewhere. However, it is not the actual case.
The output usually contains some side-lobes in addition to the main lobe. Suppose, the
ideal or desired output is denoted by D(Z) and the undesired output (having only the
side-lobes) is denoted by B(Z), such that
C(Z) = B(Z) + D(Z) (6.5)
where
B(Z) = Co + C,Z-V +0.Z-^*'+ (6.6)
and
D(Z) = 0 + 0.Z-' + . . . +CN.IZ-'^" + O.Z- ' + (6.7)
Where the length of the B(Z) and D(Z) is the same as that of C(Z) i.e. equal to (2N-1).
92
The undesired effects represented by B(Z), can now be eliminated after
subtracting it from the output C(Z), as shown in fig. 6.3 so that only the desired output is
left.
D(Z) -C(Z)-B(Z)
= A(Z) F(Z) - B(Z) (6.8)
The undesired eftects Represented by B(Z) can be generated after subtracting
D(Z) from C(Z) as shown in Fig. 6.4 i.e.
B(Z)= C(Z) - D(Z) (6.9)
The overall scheme for side-lobe elimination is shown in Fig. 6.5 where both the
matched filters are identical. As can be seen from Fig. 6.5 the output contains no side-
lobes. If the auto-correlation fiinction of the input sequence {aj} (output of the
matched filter) is normalized such that the main lobe has a value equal to unity, then the
CN-1 in equation 6.7 is equal to unity.
The over-all transfer fiinction of the scheme proposed for side-lobe elimination
can be evaluated with reference to figures 6.6 to 6.8. As can be seen from these figures
the over-all transfer fiinction of the scheme is 1/A(Z) which is same as that of an ideal
inverse fiher.
From the above discussions it can be concluded that the scheme proposed for
side-lobe elimination may be regarded as some kind of ideal inverse filtering operation,
without actually using an inverse filter.
However, the noise performance of above scheme Fig. 6.5 comes out to be
poor, because at the output of the subtracter of the upper channel both the noise terms
will be added. If the noise is white Gaussion with zero mean and variance a . Then
n - l
noise output at both the channels will be proportional to a^.X/i* = 7 Therefore after 1=0
subtracting the outputs of both the channels the noise terms will fail to cancel, therefore
overall effect of noise will increase. However, the effect of noise can be decreased
by inserting a squarer circuit in both the channels as shown in Fig. 6.9.
93
A(Z) F(Z)
C(Z)=A(Z)F(Z) •
Fig. 6:1
B(Z)
Fig . 6.3
A(Z) , , C(Z)^ B(Z)=C(Z)-D(Z)
D(Z)
Fig. 6.4
A(Z) , w r\i^}
A(Z) to VC^^ w r\i^)
-CK^)
K
-. C(T\ fc*-r •I
A(?)
-»o—^^ ) B(Z)
o DCZ)
Fig. 6.5 D(Z)
Fig. 6.6
A(Z) F(Z)-F(Z) +
A(Z)
D(Z) •
rig. 6.7
Fig. 6.8
A(Z) F(Z)
C(Z). square! po •*D(Z)
square
A(Z) F(Z)
C(Z) B(Z>
D(Z)
Fig. 6.9
The input to the squarer is C(Z) + TJ. Therefore the output will be
[C(Z) + Ti] = C'(Z) + Ti + 2TIC(Z) (6.10)
Modified B(Z) will be
B(Z) = C (Z) + r\^ + 2TIC(Z)-D(Z) (6.10)
The output of the subtracter of the upper channel will be the difference of Equ.
6.10 & 6.11 which will be equal to D(Z), which is firee fi'om undesired sidelobes and
noise.
6.4 NEURAL NETWORK FOR SIDELOBE SUPPRESSION
16.4.1 Introduction
In this section various aspects of using Artificial Neural Networks (ANN) for
sidelobe reduction are explored. After two decades of eclipse, interest in artificial neural
networks has grown rapidly over the past few years. This resurgence of interest has
been fired by both theoretical and application successes. ANN has found wide application
in pattern recognition. The pattern recognition is applicable in radar signal detection or
classification processes. In the present case Radar signals can be treated as data patterns.
The basic theories of statistical hypothesis, testing, decision theory etc. apply in these
situations. Indeed, many pattern recognition techniques employ likelihood ratios, when
the statistics are known apriori, and pattern recognition or classification is then
merely an application of multiple hypothesis testing. Nevertheless, some interesting
techniques presented under the name of pattern recognition, learning machines,
artificial intelligence etc. are applicable to radar system and should be available for the
radar designers consideration. In the following sections brief theory of neural networks
and their use for sidelobe suppression is discussed.
94
».4.2 Fundamentals of Artificial Neural Networks .
The basic implementation mechanism can be explained with reference to Figure
6.10(a). Here, a set of inputs labeled Xi, X2, . . . , Xn is applied to the artificial neuron.
These inputs, collectively referred to as the vector X, correspond to the signals into
the synopses of a biological neuron. Each signal is multiplied by an associated weight wi,
,W2, . . . , w„, before it is applied to the summation block, labelled Z. Each weight
corresponds to the 'strength' of a single biological synoptic connection. (The set of
weights is referred to collectively as the vector W). The summation block, corresponding
roughly to the biological cell body, adds all of the weighted inputs algebraically,
'. producing an output that is called NET. This may be compactly stated in vector notation
! as follows
NET = XiWi + X2W2 + . . . + xw„
=XW
Activation Functions
The NET signal is usually further processed by an activation function F to
produce the neuron's output OUT. This may be a simple linear function
OUT = K (NET)
Where K is a constant, a threshold function,
OUT = 1 , if NET > T
OUT = 0 otherwise
Where T is a constant threshold value, or a function. That more accurately simulates the
non-linear transfer characteristics of the biological neuron and permits more general
network functions.
95
NET = XW
Fig. 6.10 (a) Artificial Neuron
OUT=F(NET)
Fig. 6. IQ (b) AxtiEdal Meuron vnth Activatioa Function
Fig. 6.10 (c) Sigmoidal Logistic Function
mn
FIR 6 11 SinRlc Layer Neural Network
In figure 6.10(b) the block labeled F accepts the NET output and produces the
signal labeled OUT if the F processing block compresses the range of NET, so that OUT
never exceeds some low limits regardless of the value of NET, F is called a
'squashing function'. The squashing function is often chosen to be the logistic function or
'sigmoid' (meaning S-shaped) as shown in Fig. 6.10(c). This function is expressed
mathematically as F(x) = l/(l+e"'). Thus
OUT = 1/(1 + e' ' ')
By analogy to analog electronic systems, the activation function may be
considered as a non-linear gain for the artificial neuron. This gain is calculated by finding
the ratio of the change in OUT to a small change in NET. Thus gain is the slope of the
curve at a specific excitation level. It varies from a low value at large negative
excitations to a high value at zero excitation, and it drops back as excitation becomes
very large and positive. Grossberg (1973) found that this non-linear gain characteristic
solves the noise saturation dilemma that he posed; that is, how can the same network
handle both small and large signals, small input signals require high gain through the
network if they are to produce usable output; however, a large number of cascaded high
gain stages can saturate the output with the amplified noise (random variation) that is
present in any realizable network. Also, large input signals will saturate high gain stages,
again eliminating any usable output. The central high gain region of the logistic function
solves the problem of processing small signals, while its regions of decreasing gain at
positive and negative extremes are appropriate for large excitations. In this way, a neuron
performs with appropriate gain over a wide range of input levels.
Although a single neuron can perform certain simple pattern detection fiinctions,
the power of neural computation comes from connecting neurons into networks. The
i simplest network is a group of neurons arranged in a layer as shown in Figure 6.11. The
set of inputs X has each of its elements connected to each artificial neuron through a
separate weight.
96
Training the Network
The objective of training the network is to adjust the weights so that
application of a set of inputs produces the desired set of outputs. These input-output
sets can be referred to as vectors. Training assumes that each input vector is paired with
a target vector representing the desired output. Before starting the training process,
all of the weights must be initialized to small random numbers. This ensures that the
network is not saturated by large values of the weights and prevents certain other
training pathologies. Training' the back propagation network requires the steps that
follow:
1. Select the next training pair from the training set, apply the input vector to the
network input.
2. Calculate the output of the network.
3. Calculate the error between the network output and the desired output.
4. Adjust the weights of the network in a way that minimizes the error.
5. Repeat steps I through 4 for each vector in the training set until the error for the
entire set is acceptably low.
6.4.3 Results and Discussion
The above method is applied in the present case for suppressing the
undesired sidelobes at the output of the matched filter. Three possible schemes have
: been proposed and studied they are as follows.
Scheme 1: The neural network is used after the matched filter for further suppressing the
.response sidelobes. Neural network can suppress the sidelobe to any desired value
i (ideally zero) but as the sequence length increases number of neurons and weights also
increases making it practically difficult to realise the scheme for longer codes. For
example in the case of Barker sequence of length 13, the matched filter output (which is
also the auto-correlation fianction of the code) is of length 25. The number of weights of
97
the network becomes 25X25=625 which is quite large and obviously have practical
•limitations for implementing the scheme using appropriate hardware. As the sequence
I length increases, number of weights also increases accordingly.
Scheme II : In this scheme the matched fiher is replaced by a single neural network.
r Input is the received code sequence and output is the auto-correlation function of the
received sequence the desired output is ideal i.e. [0 0 0 0... 0 1 0 0 ... 0 0] (if the input
sequence is normalized). This scheme is used with Barker Code of length 5. The results
, are given in Table 6.4(a). Table (a) shows the sidelobe energy (SLE) and Peak sidelobe
levels with number of iterations for various value of n (the learning rate of ANN). With the
I increase in n the number of iterations are decreasing, -while SLE and PSL almost remains
constant. Table 6.4(b) shows the results when the input code sequence is of length 5 will
all element values given as+1. Table 6.4(c) shows the number of iterations with respect
' of P (Degree of non-linearity used). As p increases number of iterations decreases,
, sidelobe energy and peak sidelobe level also decreases. This table also shows the
1 variation of summation of the square of weights corresponding to the nth output, which I also decreases as P increases. Table-6.5 gives these results when the input sequence is of
length 13 (all element values +1). Table (a) shows the variation w.r.t. T and Table (b)
shows the variation w.r.t p. Fig. 6.12 shows the variation of SLE, PSL, and number of
', iteration corresponding to various values of rj & p. From Tables and Figures it is clear that
' the process is converging. However, as the sequence length increases number of weights
of the neural network also increases. When the sequence length is quite large say 100
then number of weights required are 100X199=19900 which may be quite large putting
serious limitations for practically implementing this scheme. Therefore, in order to
r reduce the number of weights on alternate scheme is being suggested and is being 1
' given as follows.
98
TABLE 6 . 4
S.NO.
1.
8
9
10
n
0.2
0.6
1.0
1.2
1.6
2.0
2.4
3.0
5.0
10
(a)
No. of
iterations
358
119
72
60
45
36
30
24
14
8
SLE
(Nor)
0.048
0.048
0.048
0.047
0.047
0.047
0.047
0.047
0.048
0.048
PSL
(Nor)
0.111
0.111
0.110
0.110
0.110
0.110
0.110
0.110
0.110
0.110
B
0.5
1.0
1.5
2.0
3.0
5.0
7.0
10
No. of
iterations
146
75
51
39
27
18
16
20
(bl
SLE
(Nor)
0.049
0.048
0.047
0.046
0.045
0.043
0.037
0.025
PSL
(Nor)
0.110
0.110
0.110
O.IIO
0.109
0.109
0.107
0.104
^ V
20.67
6.86
3.32
2.35
1.65
1.29
1.19
1.14
TABLE 6 . 5
S.No.
1
2
3
4
5
6
7
8
9
10
11
12
n
0.6
0.8
1.0
1.2
1.6
2.0
2.4
3.0
5.0
(a)
No. of
iterations
126
95
76
63
47
38
32
25
15
SLE
(Nor)
.276
.273
.273
.274
.276
.267
.267
.269
.274
1
PSL
(Hor)
0.110
0.110
0.110
0.110
0.110
0.109
0.109
0.109
0.109
^"ni '
6.40 ,
6.43
6.44
6.43
6.44
6.34
6.54
6.62
6.50
B
0.2
0.5
1.0
1.5
2.0
3.0
4.0
5.0
6.0
7.0
i 8.0
1 10.0
No. of
iterations
181
74
38
26
20
14
12
11
12
13
15
33
(bl
SLE
(Hor)
.290
.282
.271
.261
.251
.234
.195
.164
0.117
0.089
0.068
0.327
PSL
(Nor)
0.111
0.110
0.110
0.109
0.106
^V
123.30
21.26
6.49
3.72
2.73
2.02
1.78
1.66
1.57
1.57
1.54
1.53
TABLE 6.6
S.No. n No. of SLE PSL E w . ni
iterations (Nor) (Nor)
1
2
3
4
5
6
7
8
9
0 . 6
0 . 8
1 .0
1 .2
1 .6
2 . 0
2 . 4
3 . 0
5 . 0
125
93
75
62
47
37
31
25
15
0 . 0 4 8 0 . 1 1 0 6 .04
0 . 0 4 8 0 . 1 1 1 6 . 0 3
0 . 0 4 8 0 . 1 1 0 6 . 0 6
0 . 0 4 8 0 . 1 1 0 6 . 0 5
0 . 0 4 7 0 . 1 1 0 6 .10
0 . 0 4 8 0 . 1 1 1 6 . 0 8
0 . 0 4 8 0 . 1 1 0 6 . 1 1
0 .047 0 . 1 0 9 6 . 1 7
0 . 0 4 6 0 . 1 0 8 6 . 2 8
SLE = Side Lobe Energy = Total output Energy - Main Lobe Energy
PSL = Peak Side Lobe Level
E w . = Summation of the Squares of Weights Corresponding to the nth output
Tolerance = Actual output - Desired Output
n = Learning rate of the Artificial Neural Network
B = Degree of non-linearity used.
, Scheme i n : In this scheme the output of the neural network is kept as 5 in all the
• cases, independent of the length of the input code (instead of the ACF of the code as
given in Scheme II). This reduces the number of weights considerably. Table 6.6 gives
the variation of SLE, PSL, Xw ni and number of iterations for various values of n for a
code of length 13 (all element values +1). Comparing these results with the results
I obtained in Scheme II (Table 6.5(a)) it is clear that the results are almost similar except
the case of SLE which is much smaller in this case which is quite obvious as the length of
the output is reduced from 25 to 5 only. This shows that the results are better in this case,
and this achievement is obtained using much less number of weights (13X5) in this case as
compared to (13x25) used in Scheme II. Tolerance in all the cases is kept at 0.1.
6.5 SUMMARY I
If processor complexity is not of overriding concern, the use of a MMF may be I [justified in those cases where improvements in clutter performance are of significant >
magnitude. In general, there exists a trade-off between resolution and detection
performance which sets practical limits on the sidelobe suppression that can be obtained.
! However, the degradation in SNR is usually small if the input waveform is optimized for
matched conditions. Moreover*, as printed out by Rihaczek, [35] the approach of
j optimum waveform design for an MF receiver also implicitly solves the problem of
waveform design for the optimum filter in the presence of clutter. Therefore, whenever
possible it is preferable to transmit a wider spectrum to achieve the desired resolution
' rather than widening the spectrum of the receiver filter
99
CHAPTER 7
COMBINED RANGE AND RANGE RATE RESOLUTION
100
7.1 INTRODUCTION
The principal aim throughout this work has been to design waveforms for a radar
environment where the relative Doppler spread of the target is negligible. However, when
there is significant Doppler shift, the reflections fi^om a target are no longer replicas of the
transmitted waveform and the matched filter (MF) response in time and frequency has to
be considered. 1 I
The following discussions of combined range and range rate (Doppler) resolution
)f a signal is included for completeness. Detailed analysis of resolution in range and range
ate of general types of waveforms can be found in many contemporary books, [22 J, [23].
>4oreover, consideration will only be given to properties pertinent to the types of
vaveforms described in previous chapters
Although, in principle, the various methods developed to design pulse trains having
;ood range resolution can be extended to the more general case of range and Doppler
esolution, the computational efforts involved for longer sequences is quite formidable
jven for modem computers. For this reason waveform synthesis for range and velocity
esolution is commonly done by trial and judicious use of available information (e.g.
unbiguity function).
It has been shown (Chapter 2) that the range resolution property of a signal
lepends on the shape of its spectrum envelope. Based on the time frequency duality it can
)e argued, therefore, that resolution in range rate depends only on the envelope of the
lignal in the time domain. Consequently, combined range and velocity resolution depends
)n the complete waveform structure in time and frequency. Hence signals with good
•esolution in one parameter may perform very poorly when combined resolution in both
)arameters is required.
For combined resolution in range and velocity the waveform must be investigated
n terms of the complete MF response in delay and Doppler. This generalized response is
jiven by Woodward's Ambiguity Function (ABF) which has already been introduced in
chapter 2 and is repeated here for convenience
101
( ,1 )1 = Y,s(.nT+T)s\nT)e' 2xu{nT+T) (7.1)
It is noted that in the literature, the terms xli,v), | xk,V) \ and | xdx.v) | ^ are often
used synonymously for the ABF.
The ABF plays a central part in the analysis of combined resolution. This is so
because the width of the main response peak of the ABF serves as a measure for close-
target visibility in range-Dopple'r, while the low-level response and subsidiary spikes give
an indication of the self clutter and target masking problem by mutual interference. Since
the volume of the ABF over the entire (x,v)- plane is constant, the signal design problem
for combined range and velocity resolution may therefore be regarded as shifting the
unavoidable ambiguity (volume) to those parts of the (T,v)-plane where it causes least
interference for a given envirormient and application. Some of the general resolution
properties of the various types of pulse trains considered previously are discussed
subsequently using the ABF description.
7.2 AMBIGUITY FUNCTION OF PULSE TRAINS
One derivative of the ambiguity fiinction involves computing the signal out put of
the receiver filter. The ambiguity function is the complex modulation of this output signal.
The transmitted signal is
S,it) = aXOe'""^"' (7.2)
where a; = amplitude factor
u(t)= complex transmitted modulation
fo = carrier fi-equency
102
Depending on the radar cross section, range, and Doppler velocity of the target,
the return signal differs from the transmitted signal by
(i) having a different amplitude Or determined by the radar range equation.
(ii) being delayed by x and
(iii) being Doppler shifled by u
S,{t) = a^u{t-T)e'^''^''-"^'-'^ (7.3)
In the general case the receiver filter is matched to some signal Sni(t) with
aodulation u(t), and thus has the impulse response
h„(t) = KS\T,-t)
= Ka„u\T, -1 - r Je-^^'(/«-'-x^-'--) ^ ^
vhere, tm u^ are the delay and the Doppler shifts to which the filter is matched. If u(t) =
.'(t), then the filter is matched filter. If u(t) ^ v(t), the fiher is called mismatched filter. The
agnal at the fiher output in case of a matched filter is given by
ZU) = ^^">^" ^-j2T(/«-«.)r \v{t)v\t + T)e-'^'^dt .(7.5)
The quantity within brackets is known as ambiguity function
00
x(r,w)= Jv(/)v'(/ + r)e-^""'f// (7.6) —00
n discrete function, the same is given by equation 7.1
Pulse trains whose ACF decreases at a faster rate for small Doppler shifts are the
ion-linear FM type approximation. It can be seen from figures 7.1 to 7 3 and the ABF's of
hese waveforms basically exhibits the ridge lime structure which suggests a relatively
itrong range Doppler coupling. However, the linear FM property is more and more
eliminated as the order of the spectrum tapering increases.
103
For certain applications the inability to resolve targets in range and velocity along
the ridge might be unacceptable. In these circumstances a signal whose ABF approaches
that of a single strong spike (thumbtack) as shown in Figs. 7.4 and 7.8 might be adequate.
However, the expense of implementing many Doppler channels may be prohibitive.
The choice of a thumbtack ABF may be justified for high close target resolution in
the absence of any prior information of the target environment. The close-target
; resolvability is, however, achieved at the expense of introducing self clutter. Therefore, if
the target space is confined to a narrow region, there is no reason to spread the volume of
the ABF uniformly over the (x, v) plane. Moreover, if visibility of small targets is of
overriding importance, it is preferable to choose an ABF whose volume is concentrated in
strong spikes or a narrow ridge. Such an ABF trades uniform poor visibility for weak
targets (thumbtack ABF) against good visibility for most targets and extremely poor
visibility for some targets. Fig. 7.7 illustrates the relatively large increase in sidelobe levels
off the delay axes for an optimum binary sequence of length 101. It can clearly be seen that
noise-like waveforms are inherently suited to approximate thumbtack ABF's. In general,
J however, binary sequences are usually better suited to improving range resolution rather
than velocity resolution.
In summary, all of the waveforms discussed in previous chapters are optimum for
some particular clutter environment. The different pulse trains yield a wide variety of ABF
shapes. The contours may be of the diagonal ridge structure as for linear FM by signals, or
may consists of a single strong spike surrounded by a low level pedestal for noise like
waveforms, in various combination, of these basic structures. The waveforms have
different tolerances to Doppler shifts. This may be exploited for hardware savings if
ambiguity in the range-Doppler coverage can be tolerated.
Another advantage of discrete coding which has not been mentioned is the
flexibility of eliminating the range Doppler ambiguity of codes, for example, by simply
scrambling the order of the sub-pulses. Moreover, the variations possible with discrete
coded pulse trains are virtually unlimited in that the phase, amplitude, frequency and time
of transmission of each sub-pulse can be varied. The resulting multi-fiinction capability and
adaptability to a particular target environment is clearly one of the most attractive features
discrete coding.
104
(a)
i^'v
'•1' '* .d . •* .o_- . ; . . *
i j i . i j ' . i V ^ ' ••'••• • • . • : •• •;••
^ y . - . . - . - , ; ; • - ' ,'/,•
. • " . ' • • . • • - - " • • • ' > 4
128T.
.•H«.' A^-i ••-
sStt5».*i(if?f «4 ,£«ae i? -
a
c
WT
1
O
1
-1/2T
| x l t . v ) |
1/2X
(b)
*" g« 7.1 "la) ABF of 128-«lement non-llneauc FM typo puLsa . train for n •• 1 .
(b) Peak xesponso as a function of doppler s h i f t M,
UCt,v)| lb)
\ l
I
-1/2T 1/2T
'l-g* 72 I*) '^t' o^ non-ltivcar FM type pulse txaln for a -• 2 (b) Peeik response as function of doppler sh i f t v .
(a)
<',;-r :•.•:•••: .'t- -v, T V •>•; ."
. ^ • ' "
' * . .^'<'•'*•' J^^\ Lil*^-^^"*5Si.' . *• 5-'." « . * * • * • ' ' • * ^ OOfW
'^7
-1/2T
n c
iffT
3
O
1
F i g . 7.3
Ixl^v)! (b)
(a) ADK of 128-eleinont non-linear FM type pulse
txaln for n • 3
(b) Peak response as a function of dopplec shift v.
|xtT,V)| Cb)
t l
iVrvivsiUUlAViH' " %^}^^\}^^ii^Uk
'-1/2T . lyn
B.9. 7.4 M ABP of optinxn I28'«leaent bitk&ry s«qucnc« vhAa the iAitlaJ. soquaac« I s chias«n randooily.
(b) P«ak rcsponsQ as. « functloa of doppler s h i f t v.
|x(t.v)|
4 1
>w/ 4W-v.W I '
-1/2T
i\yA-M'^\l^^J^'M^
1/2T
(b)
*'iV« 7.5 (a) ABP of 100-«lcment Huffman code darlv«<l fro« a random zero pattern.
(b) Peak rospon ic as a fvmctlon of doppler shift v.
(a)
^NUdiv\/^x^^i'
IXCT,V)|
%^^SiJ^JJ!iiJJM
-1/2T
a>)
Ag« 7o («) ABT of optiJBUBt 128>ttleaent binary s«quciK:« whttu the i n i t i a l sequeace i s ctiiosen rarvdoaly.
(b) P«Ak r«spons« as « fuoctioo of dcppler s h i f t v«
E o Sw.
U>(U , 0 - 0 TJ
>s -+- ' 'Z3 ,ox . 0
E
0 0
»^ f 1
(^ u>
O-H' 0
-»-' 3 D
>•— 0
•s Q-
C <1> E a> <i>
?o i_
<1>
o o o
. 0
rr.' o d
o q d
CD q "d CM
C3 CD
d I
i- CD CD
d CD
I
( M i l
0
q d
1 1 r 11 n 11
CD
q d CM
n M 1 1 M 1
0
q d
M 1 1 1 1 M
CD 0
CD CM
1 1 I 1 1 1 1 M 1
CD CD
CD
M M
0
( I
9 T c. D
o
o c:
Q . v_ O
gs
o
o
o .q d
tro
- o o
- CJ
o - O
o .o d
O O
to o o "d CNj
o o
'9 O q d II)
I o 'd CO
I s -o o
h5 o
[-m-r O CD
d CO
11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 i T i 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 J rJ
O q d
O q d
q o o d
o q d
£ o v_ ^ 0><D o-o
• o
>> -•-'
3 O^
O O
r > «r J-
-^
O-t-' c> -»- '
J_ <V E
o
a. v_ o
v_
r r r r r r i n 11 n i n n n | 11 11 i-r-rrriT11 » TTTI i | i I i i M I TT
O O O O
o o o lO
o o o
o o lO
o o ci o
d o
o ID
d
o Ci
d C)
C.)
ri U'J
O O
- d
O CJ
d
o q d r i
O
- O
o o -d
I- J
o - - I D
(-1 (N
d U'J
E o ^
. <i>
C7>-0 O 'T>
>. - 4 - '
"u CT !o
o o
<i ^
_C
E-^ o o -+-' D O
V*—
o - f - '
o a.
C OJ
E JU (i>
r—
o • * "
^ o
o
i _
a>
( m 111TT1111111111111111 nTTri i n T i n T T rnnTTrn r m r r r 111 O q o o
o o
o CD
o
o o o
o o o' in
o q ci o
o CM
o in
o IJ
ri r J
CD O
to li 1
o Ci
}-6
CD CJ
" O i n
CD
CD
(D ID
o C~)
-cZi
o
CD
CD '"•J r-i I CD
CHAPTER ^
DISCUSSION AND
CONCLUSION
105
This thesis has reported investigations into discrete coding techniques for
I improving range resolution and clutter performance of radar systems. The waveform
considered in this work besides representing an interesting mathematical area, are
also of practical significance in related fields such as sonar, navigation, and digital
communications. The study is focused on the properties of the coded waveforms as
modulating fiinctions of a carrier signal. The design of hardware structures of radar
! processors has not been considered, since there are generally a number of ways available
to implement near optimum receivers for a given waveform. Cost, complexity and
reliability are usually the bounds set on processor design rather than physical
realizability. These problems would have to be considered for each application individually.
The waveform design approach using discrete coding offers a degree of flexibility
and has many advantages in terms of waveform shaping and processor
implementation. The variations possible with such waveform is virtually unlimited in
that the phase, amplitude, frequency and time of transmission of each sub-pulse can be
varied. This is clearly in contrast to analogue waveforms which depend on one or
j possibly two parameters. The inherent multifunction capability of discrete coded
waveforms is compatible with the requirements of modern phased array radars. In
addition amplitude and phase modulated pulse trains are particularly well suited to
digital implementation.
Throughout this work digital processing has been assumed. The appUcation of
digital processing techniques to radar becomes more practical as compactness, cheapness
' and operational speed of digital micro circuits continue to increase. Although modem
optical processing techniques sometimes provide an attractive alternative, the use of
digital method with its inherent flexibility and reliability offers many advantages. To
I mention but a few, it simplifies pulse compression and real time multi-dimensional analysis
of input data in range, Doppler, bearing, etc. Furthermore, it also offers considerable
106
advantages in post detection and display processing. In addition the use of digital
processor will in many cases reduce future system modifications to easy and inexpensive
software changes, rather than requiring costly hardware replacements.
An attempt has been made in this work to present results of various design
>bjectives. The assumption of a matched filter receiver underlying most of the work, is not
I serious limitations on the applicability of the results, since in practice very little prior
nformation about the target environment is usually available. Therefore, the investigations
vere concentrated on the auto-correlation fiinction properties of the various types of
>ulse trains considered. The principal aim has been to design pulse trains whose auto-
lorrelation side lobes are as low as possible. The design methods presented in this thesis
lave shown that phase coded pulse trains can improve the range, resolution and clutter
ejection performance of a radar system. Self clutter interference introduced by
nadequacies in the matched filter response impose rather fiindamental limitations on weak
arget visibility. Moreover, the resolution problems caused by self clutter and undesired
)bjects are in principle no different in that both impede resolution in the same manner. If
)rocessor complexity is not of utmost concern, the use of amplitude and phase modulated
)ulse trmns may offer an improved performance. Alternatively, self clutter suppression
nay be achieved by side lobe reduction filters or methods proposed in this thesis.
\lthough mismatching the receiver filter may be useful means of adopting the
vaveforms to a particular target environment, it should not be used as a primary method
0 improve resolution. Hence, the proper approach to clutter suppression, except in few
special cases, is via waveform design and matched filtering.
In situations where the relative Doppler spread of the targets can not be
leglected, the matched filter response has to be investigated in terms of the ambiguity
function. In principle, the methods developed could be applied to the more general
problem of designing pulse trains suitable for resolving targets in range and range rate.
However, even with modem computers, the search for phase codes having good resolution
sroperties in both range and range rate is so expensive that it would have to be
107
restricted to relatively short sequence and small areas of the range Doppler plan. For
, these reasons waveform synthesis for range and range rate resolution usually consists of
' a trial and error procedure and a judicious use of available information.
In summary, the signal problem has in general defied solution by all means
other than exhaustion. In particular, no concise set of necessary and sufficient
; conditions has been formulated by which signals with specified properties can be
synthesized. Signal theory, the basis for many technical advances is far from being
complete and its fijrther development is, therefore, of fundamental importance. As a
final remark the design methods developed in this thesis are of general interest. With
appropriate modifications they can be applied to a variety of signal and filtering problems.
108
APPENDIX-A
RECURSIVE SOLUTION OF SIMULTANEOUS
EQUATIONS INVOLVING AN
AUTOCORRELATION MATRIX
109
It is possible to take advantage of the special form of an auto-correlation matrix
in order to reduce the computational work figured for the solution of a set of
simultaneous equations involving such a matrix. More specifically, an auto-correlation
matrix, say
R.= 1 'i
.r. Vl
'n-l
;has the property that all the elements on any given diagonal are the same. For example,
XQ is the element which appears on the main diagonal; ri is the element which appears
on the first super-diagonal as well as on the first sub-diagonal; etc. Hence the entire
;(N+l)x(n+l) auto-correlation matrix R„ does not involve (n+1)^ distinct elements, but only
n+1 distinct elements, namely ro, ri, X2, . . ., rn. .Let us now see how we can take
advantage of this structure.
Suppose that we wish to solve the set of simultaneous equations given by
fo ro + f) ri + . . . + fm rm = go
fo ri + fi ro + . . . + f„ Fm-i = gi (1)
fo rm + fl Tm-l + . . . + fm fo " gm
tfVhere the auto-correlation coefficients ro, r i , . . . , fm as the right hand side coefficients
Jo, gi, ..•, gm represent the known quantities, and the filter coefficients U,
represent the unknown quantities. The structure of the auto-correlation matrix makes
[possible the following recursive method for the solution of these equations.
The recursive procedure solves the equations in a step-wise manner. The step n =
3 is given as an initial condition, and then the steps n = 1, n = 2, . . ., n = m are done
successively in a recursive manner. The desired filter coefficients are the ones which
110
result on the completion of the final step, namely step n = m. A description of one of
iJHhese steps provides the necessary information to enable one to program the method for
digital computer. Let us suppose therefore that we have completed step n, and we wish to
do step n+ 1.
The completion of step k = n requires that we have computed and have retained in
I machine storage the numerical values of the following quantities:
3iiO, 3iil. • • •» Snni Otn>Pn
)i
and
InO» Inl> • • • J Inn;Yn
By definition, these quantities satisfy the matrix equations :
(ano, 0) R„.i = (On, 0 , . . . , 0, p„) (2)
id
(fnO, fnl, . . . , fm,, 0 ) R„+i = (go, g l , . . . , g„, Yn) ( 3 )
here the parentheses enclose lx(n+2) row vectors and where R„+i is the (n+2) x (n+2)
ito-correlation matrix.
Because of the special structure of the auto-correlation matrix, equation (2)
ay be manipulated into the equivalent from given by
(0, a™,..., a„i, a„o) Rn+i = (Pn, 0 , . . . , 0, a„) (4)
Let us now multiply equation (4) by a constant k„, as yet undetermined and
len add the result to equation (2). We obtain
= (a„ + k„B„,0,...,0, p„ + k„yn) (5)
111
,.|We want equation (5) to be identical to the equation
•V.'
(a„-i, 0,
= (a„.i, 0 , . . . , 0, 0) (6)
In order for equations (5) and (6) to be identical we must first of all require that
This equation allows us to determine the constant kn, that is, we compute kn by
the formula
k„ = -B„/a„ (7)
The identity of equations (5) and (6), together with our knowledge of kn,
then allows us to compute the quantities
an+1,0 ~ anO
an+1.1 = 3,1,1+ k„ann (8)
3n+I, n ~ an n "> Knanl
3n+l,n+l ~ knanO
i as well as
a„+i = a„ + k„Pn (9)
If we define the polynomials An(z) and An+i(z) as
r
A„(z) = a„o + a„iZ + . . . + a Z"
I and
An+2(Z) - an+1,0 + a„+l , lZ + . . . + anfl,i,Z"+an+l,n+jZ'
i ,
112
n+1
then we see that the equations appearing in (8) may be compassed within the confines of
one equation, namely
A„.i(Z) = A„(Z) + k„Z(a™ + . . . + a„,Z"-* + a^")
which is 1
a„.,(Z) = A„(Z) + k„Z"''A„(l/Z) (10)
This equation demonstrates the recursion from the known polynomial Ao(Z) to
le unknown polynomial A„+i(Z).
Next, if we compute
Pn+i =.an+i,o rn+2 + a„+i,.rn+i + . . . + an+u+i ri (11)
len we will have computed all the quantities in the equation
(an+1,0, an+1,1, . . .,an+l,n+l,o) Rn+2 = (Otn+1,0, . . . , 0 , P n + l ) ( 1 2 )
vhere the parentheses enclose lxn+3) row vectors and where Rn+2 is the (n+3)X(n+3)
Lito-correlation matrix. Equations (2) and (12) are counterparts; equation (2) pertains to
:ep n whereas equation (12) pertains to step n+1.
Because of the structure of the auto-correlation matrix, equation (6) is the same
5 the equation
(an+i,n+i, an+i,n, •. •, an+i,a„+i,o) Rn+1 = (0,0, —,0,a„+i) (13)
Let us mukiply equation (13) by a constant q„, as yet undetermined, and then add
le result to equation (3). We obtain
(fiiO"'"qn3n+l4i+l>Inl''"qn3n+l^, • • •, fnn"''qnan+l,l,qnan+l,o) Rn+1
= (go, g i , . . . . gn, Yn + qna„+i) (14)
^e want equation (14) to be identical to the equation
(fn+l,0,fn+J,l, . . . , fn+U,fn+l,n+l) Rn+1 = ( g o , g l , gn,qn+l) ( 1 5 )
113
In order for equations (14) and (15) to be identical we must first of all require that [!•
I
Yn + q n a n 4 l = g „ + l
I This equation allows us to determine the constant qn; that is, we compute qn by
the formula
t
qn = (gn+i - Yn) / a„+i (16)
The identity of equations (14) and (15), together with ourlcnowledgeofqn, allows us
I to compute the quantities
fn+1,0 = fnO + qn qn+l,n+l
f n + l , l = f n l + qnan+l^.. . .- . ( 1 7 )
In+l^+l ~ qn an+1,0-
[f we define the polynomials Fn(z) and Fn+l(z) as
F n ( Z ) = fnO + fn lZ + . . . + fnnZ
uid
F n + l ( Z ) = fn+1,0 + fn+l,lZ + . . . + fn+l.nZ +fn+l,n+lZ
lien the equations appearing in (17) may be encompassed within the confines of one
equation, namely
F„.,(Z) = F„(Z) + q„Z„+,A„., (1/Z) (18)
This equation demonstrates the recursion fi^om the known polynomials Fn(Z)
uid A„+i(Z) to the unknown polynomial Fn+i(Z).
Next if we compute
Yn+l - fnM.O rn+2 + fn+l,irn+l + . . . + fnH,n+iri ( 1 9 )
114
; iljthen we will have computed all the quantities in the equation
,In+l,l • • • In+M+lj 0) R„+2 = (go, gl . • . gn+1, Yn+l) (20) .1
Where the parentheses enclose lX(n+3) row vectors and where Rn+2 is the (n+3)X(n+3)
auto-correlation matrix. Equations (3) and (20) are counterparts; equation (3) pertains to
stepn and equation (20) pertains to step n+1. We have thus completed our task; we
have done step n+1 given that we had step n.
Let us now summarise the computations required. We are given the quantities.
roj r i , . . •, rm
go, gl, • • •, gm-
the completion of step n we have the quantities
anO, anl, • • • , ann, Otn, Pn
InO, Inl • • • , Inn, Yn
machine storage. To obtain step n+1 we do the following computations. We
npute kn by means of equation (7). Then we compute
an+1,0, an+1,1, . . . an+l,n, an+lji+1, Otn+i, pn+l
means of equations (8), (9) and (11). We compute qn by means of equation (16). Then
compute
tnH.O, fnU.l , • • • , ln+l,n, fn+l,n+l, Yn+1
means of equations (17) and (19). This completes step n+1.
To obtain the final result, namely the solution of the simultaneous equations
we must start at step n = 0, that is start with initial quantities which we take as
115
.,,, aoo = a, do = To, Po = fi, 5)0 - go/ro, Yo - ^ fj
Then we do the recursions for n = 1 to n = m; the final values obtained for the fiher
coefiicients, namely
lm,0» Im,l) • • • > Ii 'iii,in
represent the solution fo,fi,..., fm of the normal equations (1). Instead of having the
value of m fixed in advance, it is also possible to monitor the recursions according to the
value of the error associated with the filter at each step of the recursions and the
recursions can be stopped when this error reaches a certain limit.
The machine time required to solve the simultaneous equations (1) for a filter
with m coefficients is proportional to m for the recursive method, as compared to m for
conventional methods of solving simultaneous equations. Another advantage of using
this recursive method is that it only required computer storage space proportional to m,
' rather than m as in the case of the conventional methods.
This recursive method can also be extended to the matrix valued case in order to
compute the coefficients of multi-charmel filters. Also there is a sideways recursion which
' corresponds to shifting the time index of the right-hand side of the simultaneous
' equations. This side ways recursion is valuable in applications for it represents a relatively
index-pensive way of determining the filter that incorporates the optimum time-delay
I between the input signal and the desired output signal. For example, the sideways
recursion can be used to determine the optimum spike position for a spiking filter and
the optimum positioning of the desired output for a shaping filter.
116
APPENDIX-B
LIST OF SEQUENCES
117
B.l Binary Sequences (+= 1,- = -1)
N Sequence
'17 _ -I _ _ )- - 4 _ I t -I- f + . . + f
19 + + + . + + + . + + - + + . + -
31 . + - + + - + - - + + - + + - - + + + + + + + + + +
41 + . - + + + - - + + + - . + - - + . - + - + - + - + - + + - + + + + + + +
61 -_ + + - + - - + + + - - + + + - - + - - + - - + - + - + - + - + + - + + + + + + +
+ + + + + - + + + -+"+--
91 . + _. + + + 4. .4 ._.4. + + . + + + + + + + + + ++ + _ + + + _ + +
+ . - + + . + + + . + + . + + + - - + + + - - + - - + + - + - - + + + - - + + + - - + - -
+ - - + - + _ -_ + - + + - + - + -
101 + + + + + + + + + + + + + + + + -• + + + + + + + . . + + + - - +
+ + + - + + - - + + - - + - - + - - + - - + - - + - - + - + - - + - + - - + - + - + + -
128 + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ -- + + + -- + + + -- + + -- + + - - + + -- + + -- + -- + + - f + - + 4-- + + - +
+ - + - + + - + -- + - + + - + - + - + -- + - + - + - + - + + - +
118
105
253 + + + - + - + - + + + + + + + + + + + + + + + + -4-
-- + + + + + + + + . + + + + + + + + + + +
+ + + . . + + + + - - + + + + + ._ + + . . + + + . . + + - . + + -_ + + -- +
+ __ + + - + + - + + -- + -- + + - + + - + + .- + -- + + - + + - + + - + + - + -- + -
- + _. + _ + + + . + + . + - + + _ + - + + , + - + + - + - + -- + - + - + - + + -
+ + + + + - + - + - + + + '-- + + - + + + + + + - . + + + - + + -- + +
+ + + + + + + - + _ + + + 4- + . + + + + + + +
+ + + - + + + - + - + + + + -- .. + - + + .. + + + + + + + + + - +
- + -- + + + - + - + - + -- + - . + + + + + -- + + -- + - + +-- + + - + -- + +-
+ + + + '+4- + + - + + + -- + - + + + + - + -- + - + - + + -- + -- + -
119
B-2 Set of Sequences
N Sequence
19 + . . + . + + . + + + - + + +
19 + + . - + + + + + + 4-- + - - + + - +
19 + + + + - - + - - + - + -
31 +- + + + + + + + -- + -- + - - + - + - + -
31 _ + + + - + + + + + . + - + + - + -•---
31 + + + + + . - + + + - . + - . + . . + - + - + - +
53 - - + + + . - + + + + - . + . . + + _4- + - + - + - - + + - + + + - . + - +
53 + + + + - + - + - - + - + - - + + - + - + - + - + + + . - + + + + + - . + + +
53 + + - + - - + + + - - + + + - - + - - + - - + - + - + + + - + + - + + + + + + + +
73 + + __ + + _4._ + ._ + _. + . + 4.. + . + __ + + . + + _ + + . . + . + + + _
+ + + + - + + + + + + - . + + + . + + ._
73 +_ + _ + + + -|. + + + -|. + __-|.._ + + _.4. + + 4. + + - + + -
- - + + + . + + . . + . + + + _
73 + - + + + + - + + + + - + + - + + + - + + + - - + + ._ + + - + - - +
+ + - - + + + - - + - - + - - + - + - + - + - - + -
91 . + + - + _. + . . + . + + . - + + - - + -4- + - + - + - - + + - + + + - - + + - + +
+ __ + + + + _ + + 4. + + 4._ + ^ + 4.4.^._. + + -|.4.
91 ._ + + - - + - + + + . + + + + _- + - + + + - + + + + - + - + + + + - + - - + + + - + -
- + - + + + — + . + - + . + - - + - + + . + + _ + + + - + + + —
120
91 + - + - + + _ + _ + . + . + . . + _ . + _ . + + + . . + + + - - + - + + - - + +
+ + + _ + -t. + _4.+-f + + + 4. + 4. + + 4._ + 4.4.__ + _4.4. + + . _ 4 . _ .
1 0 1 f - - l I - - I I I I l - l I l - t l - l l - l l - l l - l | _ _ | . | - | I I -
- + - + - + - + - + + + + + + + + - + + + + + + + + + + + +
+ _ . + + + - - + +
101 - + 4-- + + + + + + - + - + + - + + + - + - + + - - + - + - + + - + +
+ + + - - + + + - . + ' - - + + + . + + + . + + - + _ + + + - - + -
+ + - - + + -
101 _ + - + - + - + - + + - + + - + + - + . . + , . + . . + - . + + _ + -_ + + ._ +
+ - + + + + __ + + + _ ,4- + + + + + + 4.4. + 4. + + + + + +
+ + . + + +
121
B-3 Multilevel Sequences *N Sequence
10, 1081, 0597, 079, 0686, -0905, -1086, 0905, 0686, -0791, 0597,
-1 08, 1 0
0 139, 0 160, -0034, -0 097, 0 045, 0 110, 0 098, 0 131, 0 224, 0217, -
13
31
61
91
0 187
0 205
-0 066
0 023
0 070
0 071
0 156
0 071
0 070
0 035
0 116
0 108
0 040
0 040
0 108
0 116
0 035
-0 278, 0 359, 0 323, -0 047, 0 251, -0 037, -0 137, 0 200, -0 234, -
-0 134, -0 171, 0 175, -0 186, 0 232, -0 123. 0 196, 0 013, 0 097,
-0 041, 0 068, -0 105, 0 151, -0 197, 0 227, -0 223, 0 167, -0 060, -
0 168, -0 175, 0 075, 0 078, -0 170, 0 119, 0 042, -0 161, 0 106,
-0 161, 0 040, 0 136, -0 114, -0 084, 0 146, 0 039, -0 155, -0 015,
0 015, -0 155, -0 039, 0 146, 0 084, -0 114, -0 136, 0 040, 0 161,
-0 106, -0 161, -0 042, 0 119, 0 170, 0 078, -0 075, 0 175, -0 168, -
0 060, 0 167, 0 223, 0 227, 0 197, 0 151, 0 105, 0 068, 0 041, 0 023,
-0 027, 0 044, -0 067, 0 096, -0 130, 0 164, -0 190, 0 197, -0 176, 0 120, -
-0 060, 0 135, -0 157, 0 110, -0 010, -0 096, Q 145, -0 103, -0 010,
-0 131, 0 037, 0 089, -0 133, 0 047, 0 087, -0 127, 0 025, 0 107, -
-0 027, 0 127, -0 053, -0 096, 0 105, 0 045, -0 124, 0 004, 0 123, -
-0 113, 0 062, -0 104, -0 072, -0 101, 0 072, 0 104, 0 062, -0 113,
0 123, -0 004, -0 124, -0 045, 0 105, 0 096, -0 053, -0 127, -0 027,
0 107, -0 025, -0 127, -0 087, 0 047, 0 133, 0 089, -0 037, -0 131, -
-0 010,0 103, 0 145, 0 096, -0 010, -0 110, -0 157, -0 135, -0 060,
0 120, 0 176, 0 197, 0 190, 0 164, 0 130, 0 096, 0 067, 0 044, 0 027,
122
REFERENCES 1. RJHACZEK, "Radar signal design for target resolution' Proc IpEE, Vol 53, pp 116-128,
February 1965
2. A W RIHACZEK, 'Radar resolution properties of pulse trams' Proc IEEE, Vol 52, pp 153-164, February 1964
3. D E HUNT,'Improved multifunction array radar pefionaax^ with multifrequency operations (with technology prediction for 1978)' EASCON Conv Record, pp 52-59, September 1968
4. SIEBERT, 'A radar detection philosophy', IRE Trans Inform Theory Vol IT-2, pp 204-221, September 1956
5. NZIERLER, 'Linear recumng sequences', J SIAM APPL Math, Vol 7, pp 31-48, March 1959
6. R M LERNER,'Signals with umform ambiguity function*', IRE Trans Nat Conv Record, pt 4, pp 27-36, March 1958
7. J E STORER and R TURYN, 'Optimum finite code groups', Proc IRE (Corr ), Vol 46, p 1649, September 1958 •
8. R H BARKER,'Group synchronizing of binary digital systems', in 'Communication theory', W Jackson (ED ) pp 273-287, Butterworths, 1953
9. R J TURYN, 'On Barker codes of even length', Proc IEEE (Corr), Vol 51, p 1255, September 1963
10. A M BOEHMER, 'Binary pulse compression codes', IEEE Trans Inform Theory, Vol IT-13, pp 156-167, April 1967
11. DEVAKMAN, PNSEDLETSKIY and I Z SHAPIRO, 'Synthesis of quantized phase-anipulated signals with good autocorrelation properties'. Radio Eng Electron Phys (USSR), Vol 15, pp 606-614, Apnl 1970
12. LYEVARAKIN, 'Synthesis of phase-manipulated $i^ials'. Radio En Electron Phys (USSR), Vol 14, pp 690-698, May 1969
13. R TURYN and STORER, 'On bmary sequences', Proc Am Math Soc , Vol 12, pp 394-399, June 1961
14. R TURYN, 'Sequences with small correlation', in 'Error correcting codes', H B Mann (Ed), pp 195-228, John Wiley 1968
15. E HOLLIS, 'Comparison of combined Barker codes for coded radar use', IEEE Trans Aerospace and Electron Sys , Vol AES-3, pp 141-143, January 1967
16. R C HIEMILLER, 'Phase codes with good periodic oorrelation properties', IRE Trans Inform Theory, Vol IT7, pp 254-257, October 1961
17. RL FRANK, 'Polyphase codes with good nonperiodic correlation', IRE Trans Inform Theory, Vol IT-9, pp 4345, January 1963
18. RL FRANK and SA ZADOFF, 'Phase shift pulse co<|es with good periodic correlation properties', IRE Trans Inform Theory, Vol IT-8, pp 381-382, October 1962
19. S W GOLOMB and R A SCHOLTZ, 'Generalized Barker sequence', IRE Trans hiform Theory, Vol IT-11, pp 533-537, October 1965
123
20. D.A.HUFFMAN, 'The generation of impulse equivalent pulse trains', IRE Trans. Inform. Theory, Vol. IT-8, pp S10-S16, September 1962.
21. P.M.WOODWARD, 'Probability and information themy with applications to radar', Pergamon Press, 1955.
22. C.E.Cook and M. BERNFELD, 'Radar Signals', Academic Press, 1967.
23. A.W.RIHACZEK, 'Principles of high-resolution radar', McGraw-Hill, 1969.
24. R.PRICE and E.M. HOFSTETTER, 'Bounds on the v«lumo and height distributions of the ambiguity function', IEEE Trans. Inform. Theory, VQI.IT-1 1, pp 207-214, April, 1965.
25. J.R.KLAUDER, AC. PRICE, S. DARLINGTON and W. ALBERSHEIM, 'The theory and design of chirp radars', Bell Sys. Tech. J., Vol. 39, pp 745-808, July 1960.
26. M.I.SKOLNIK, 'Introduction to radar systems', McGraw-Hill, 1962.
27. M.H.ACROYD, 'Amplitude and Phase modulated pulse trains for radar. The Radio and Electronic Engineer, Vol. 41, pp 541552, December, 1971.
28. L.E.FRANKS, 'Signal theory'. Prentice Hall, 1969.
29. J.DUGUNDJI, 'Envelopes and pre-envelopes of real wav^orms', IRE Trans. Inform. Theory, Vol. IT-4, pp 53-57, March, 1958.
30. C.W.HELSTROM,'Statistical theory of signal detectioq', (2nd Ed.), Pergamon Press, 1968.
3L L.R.RABINER and B.GOLD, 'Theory and Application (rfdigittl signal processing'. Prentice Hall, 1975.
32. J.W.COOLEY and J.W.TUKEY, 'An algorithm for macjiine calculation of complex Fourier series'. Math. Comput., Vol. 19, pp 297-301, ^ r i i , 1965.
33. Special issue on the fast Fourier transform, IEEE Traas. Audio and Electroacoustics, Vol. AU-15, June, 1967.
34. W.B.DAVENPORT and W.L. ROOT, 'An introduction to tiie |heory of random signals and noise', McGraw-Hill, 1968.
35. A.W.RIHACZEK, 'Optimum filters for signal detection in clutter', IEEE Trans. Aerospace and Electron. Sys., Vol. AES-1, pp 297-299, December, 1965.
36. L.J.SPAFFORD, 'Optimum radar signal processing in clutter', IEEE Trans.Inform. Theory, Vol. IT-14, pp 734-743, September, 1968.
37. D.F.DELONG (Jr.) and E.M.HOFSTETTER,'On the design qf optimum radar waveforms for clutter rejection', IEEE Trans. Inform. Theory, Vol. lT-13, pp 454-463, July, 1967.
38. W.D.RUMMLER, 'Clutter suppression by complex wes t ing of coherent pulse trains', IEEE Trans. Aerospace and Electron. Sys., Vol. AES-2, ^ 689-699, >3ovember, 1966.
,,,39. C.A.STUTT and L.J. SPAFFORD,'A best mismatd»ed filter response for radar clutter discrimination', IEEE Trans. Inform. Theory, Vol. IT-14, pp 280-287, March, 1968.
40. J.SAKAMOTO ct al.,'Coded pulse radar system', J. Faculty Eng. Univ Tokyo, Vol. 27, pp 119-181, 1964.
41. P.E.BLANKENSHIP and EM. HOFSTETTER, 'Distal pulse compression via fast convohition', IHEE Trans. Acoustics, Speech and Signal Processing, Vol. ASSP-23, pp 189-201, April, 1975.
124
42. D.E.VAKMAN, 'Sophisticated signals and the uncertainty principle in radar', Springer-Verlag, 1968.
43. E.A.ROBINSON, 'Statistical communication and detection with special reference to digital data processing of radar and seismic signals'. Griffin, )%7.
44. J.H.MILLER,'On the location of zeros of certain classes of polynomials with ^ iplications to numerical analysis', J. Inst. Maths. Applies., Vol. 8, pp 397-406, 1971.
45. H.B.VOELCKER, Toward a unified theory of moduUtfion', Proc. IEEE, Vol. 54, pp 340-353, and pp 735-755, 1966.
46. M.H.ACKROYD, 'Computing the coefficients of higli-order polynomials', Electron. Letters, Vol. 6, pp 715-717, October, 1970.
47. E.L.KEY, E.N.FOWLE and R.D. HAGGERTY, 'A method of designing signals of large time-bandwidth product', IRE Int. Conv. Record, pt. 4, pp 146 -155, 1961.
48. E.N.FOWLE,' The design of FM pulse compression signals', IEEE Trans. Inform. Theory, Vol. IT-10, pp 61-67, January, 1964.
49. M.BERNFELD,C.E.COOK, J.PAOILLO and C.A. PALMIERI, 'Matched filtering, pulse compression and wavefo.rm design (four parts), Microwavo J., pt. 1, pp 56-64, October, 1964; pt. 4, pp 73-81, January, 1965.
50. D.C.CHU, 'Polyphase codes witli good periodic properties',
51. IEEE Trans. Inform. Theory, (Corr.), Vol. IT-18, pp 531-532, July, 1972.
52. R.ROY and O.LOWENSCHUSS, 'Chirp waveform gcperation using digital samples', IEEE Trans. Aerospace and Electron. Sys., Vol. AES-10, pp 451-436, December, 1970.
53. L.I.BLUSTEIN, 'A linear filtering approach to the computation of discrete Fourier transform',IEEE Trans. Audio and Electro-acoustrics, Vol.AU»18, pp451-456, Dec- 1970.
54. L.W.MARTIONSON and R.J. SMITH, 'Digital matched filtering with pipelined floating print fast Fourier transform', IEEE Trans. Acoustic, ! )eech and signal processing. Vol. ASSP-23, pp 222-234, April, 1975.
55. P.V.INDIRESAN and G.K. UTTARADHI, 'Iterative method for obtainmg good aperiodic binary sequences',J.of Optimization Theory and Appl.,V.7,pp 90-l08,Feb- 1971.
56. B.LIU and A. PELED, 'A new hardware realization of high speed fast Fourier transformer', IEEE Trans. Acoustic, speech and signal processing. Vol. ASSP-23, pp 543-547, December, 1975.
57. J.LINDER, 'Binary sequences up to length 40 with \>est possible autocorrelation fiinction'. Electron. Letters, Vol. 11, p 507, October, 1975.
58. P.S.MOHARIR,'Temary Barker codes'. Electron. Letters, Vol. 10, pp 460-461,Oct., 1974.
59. M.J.E.GOLAY, 'Complementary Series', IRE Trans. Information Theory, Vol. IT-7, pp 82-87, April, 1961.
60. G.R.WELTI, 'Quaternary codes for pulsed radar', IRE Trans. Inform. Thoery, Vol. lT-6, pp ; 400-408, June 1960.
6L C.C.TSENG & C.L.LIU, 'Complementary sets of soquence'.IRE Trans. Inform. Theory, Vol. IT-18, pp 644-652, September, 1972.
62. W.SCHREMP and T.SEKIMOTO, 'Unique word detection in digital burst comm.', IEEE Trans. Communications Technology, Vol. COM-16, pp 567-605, August, 1968.
125
63. E.A.ROBINSON, 'Multichannel time series analysis with digital computer programmes', (Holden Day, 1967).
i! 64. N.B.CHAKRABARTI, 'Estimate of a bound for the miairauni value of the cross-conelation
between two binary* sequences'. Electron. Letters, Vol. 6, pp 686-687, October, 1970.
65. D.M.HIMGELBLAU, 'Applied non-linear programmi^', Mc Graw Hill, 1972.
66. J.W.BANDLER and C.CHARALAMBOUS, 'Practical leasf pth optimization of network', IEEE-Trans. Microwave Theory and Techniques, Vol. M|T-20, pp 834-840, December, 1972.
"67. A.G.DECZKY,'Syntliesis of recursive digital filters using the minimum p-error criterion', IEEE Trans. Audio and Electroacoustics, Vol. AU-20, pp 257-263, Oct-1972.
68. H.EKBLOM and S. HENSIKSON, 'Lp-criteria for tfae estiniation of location parameters', SIAM J. Appl. Madi., Vol. 17, pp 1130-1141, November, 1961.
69. M.J.D.POWELL, 'Finding minima of functions oi several variables', in "Numerical analysis', J. Walsh (Ed.), pp 143157, Academic Press, 1966.
70. D.J.WILDE, 'Optimum seeking methods', Prantice-Hall, 1964.
71. M.J.D.POWELL, 'A method for minimizing a sum of squares of non-linear flinctions without calculating derivatives', Comput. J., Vol. 7, pp 303-307,1965.
72. R.FLETCHERj'Generalized inverse methods for the beat leait squares solution of systems of non-linear equations', Comput. J., Vol. 10, pp 392-399, 1968.
73. M.J.D.POWELL, 'A Fortran subroutine for solving systems of non-linear algebraic equations', Harwell Report, AERE-R5947, Nov. 1968.
"74. U.SOMAINI,'Binary sequences with good correlation prcyerties', Electronic Letters, Vol. 11, June, 1975.
75. J.R.CAPRIO, 'Strictly complex impuse-equivalent codes apd subsets with very uniform amplitude distributions', IEEE Trans. Inform. Theory, Vol. lT-15, pp 695-706.
76. M.H.ACKROYD, 'The design of Huffman scquofices', |EEE Trans. Aerospace and Electron. Sys., Vol. AES-6, pp 790-796, November. 197Q.
'Ii77. E.L.KEY,E.N.FOWLE and R.D.HAGGERTY, 'A methqd of sidelobe suppression in phase-coded pulse compression systems', M.I.T. Lincoln Lab., Lexington, Tech. Rep. 209, August, 1959.
78. E.A.ROBINSON and S. TREITEL, 'Principles of digital Wiener filtering', Geophys. Prosp., Vol. 15, pp 311-333, 1967.
79. M.H.ACKROYD and F. GHANI, 'Optimum mismatched fijters for sidelobe suppression', lu IEEE Trans. Aerospace and Electron. Sys., Vol. AES-9, pp 214-218, March, 1973.
' i l '
80. ACKROYD, 'Huffinan sequences with approximately uniform envelopes or cross-correlation functions', IEEE Trans. IT. Vol-23, p 620-^23, 1977.
81. GOLD, 'Optimal binary sequences for spread spectrum multiplexing', IEEE Trans. IT.-13, p6|i)-62l, 1967.
82. HUFFMAN, 'A modification of Huffman, Impulse equivalent pulse Trains to increase signal energy utilization', IEEE Trans. IT-20, p 559-561, 1974.
'"83. MASSEY and J.J. UHRAN, Jr., 'Sub band coding; in Proc. 13th Ann. Allert on Conf. Circuit and System Theory, p 539547, 1975.
126
84. PURSLEY and H F A ROEF, 'Numerical evaluation of correlation parameters for optimal phases of bmary shift register sequence', IEEE Tran$. Comm Vol Comm 27, p 15971604, 1979
85. PURSLEY and D V SARWATE, 'Bounds on apertotUc crosscorrelation for binary , sequences'. Electron Letter, Vol 12, p 304-305, 1976.
*'86. PURSLEY and D V SARWATE, 'Evaluation of correlntion parameter for periodic sequences', IEEE Trans IT, Vol 23, p 508-513, 1977
87. PURSLEY and D V. SARWATE, 'Performance evaJuation of phase coded spread spectrum multiple across comm part II code sequence analyf is, IEEE Trans Comm, Vol COM-25, p 800-803, 1977
88. SARWATE, 'Bounds on cross correlation and auto-corndatioq of sequences', IEEE's Trans IT 25, p 720-724, 1979
89. SHEDDandDV SARWATE,'Construction of sequence wiftgood correlation properties', IEEE Trans IT-25, p 94-97, 1979
90. SARWATE and M B PURSLEY, 'Cross correlatiOB properties of psendorandom and related sequences', IEEE, Proc 68, p 593-619, 1980
91. LEWIS and F F KRETSCHNER, 'A new class of pcrfyphase pulse compression codes and .|, techniques', IEEE Trans , AES17, p 364-372, 1981 ' ( •
92. KERDOCK, R MAYER and D BOSS, 'Longest binaiy pulse compression codes with given peak side lobe levels Proc IEEE-74, p 366, 19^6
93. RAO and V U REDDY, 'Biphase sequence generation with |ow side lobe auto-correlation function', IEEE Trans AES22, p 128-133, 1986
94. TITLLEBAMM, Time frequency hop signals part I, ootbng based upon the theory of Imear congrvcnccs', IEEE Trans AES-17, 1981
'''95. QUYNH and S PRASAD, "New class of sequence sets with good auto and cross-correlation function', Proc IEEE, Vol 133, part F, 1986
96. ZORASTER, 'Minimum peak range side lobe filter for biijary phase coded waveforms', IEEEAES-16,p 112-5, 1980
97. MESSE and D GUILI, 'Range side lobe elimination ui please coded pulse radars', Proc IEE-129, part F, p 289-96, 1982
••'•98. TAYLOR and H J BLINDRIKOFF, 'Quadriphase code - A radar pulse compression signal with unique characteristics', lEEE-AES, Vol 24, March, 1988
99. Z A ABBASI and F GHANI, 'Multilevel sequences with good auto-correlation properties'. Elect Lett 24, p , March, 1988
100. Z A ABBASI, 'Iterative method for range side lobe suppression for binary codes'. Elect L c U , J i i l> , l')XR
;|101. BICOCHI, T BUCCIARELLI, PT MELACI, 'Radar sensitivity and resolution m presence of range side lobe reducmg networks designed using Imear programming' The Radio and Electronic Engineer, Vol 54, No 6, June, 1984
102. HUNT and MH ACKROYD, 'Some mteger Huffinan sequences'. Trans IEEE, Vol IT-26, 1980
103. RUTTER and P M GRANT, 'Generation of codes w»th good autocorrelation properties'. Elect Lett, Vol 19, 1983
127
104. MOHARIR, K. VENKATA RAO and S.K. VERMA, 'Monogenic function range resolution radar, Proc. lEE, Pt. F, Vol. 134, 1987.
105. HOHOLDT and J. JUSTASEN, 'Determination of the merit factor of Legendre sequences'. Trans. IEEE, Vol. IT-34, 1988.
106. SOLE, 'A quaternary cyclic code and a family fo quadri phase sequences with low correlation properties'. Coding Theory and AppUcatiop, Lecture notes in Computer Science, New York, Springer Verlag, 1989.
107. MOHARIR, "New figures of merit for pulse compression sequences', J. lETE, Vol. 38, No. , 4, 1992.
108. LUKE, 'Families of polyphase sequences with near optimal two valued auto and cross-correlation function'. Electronic Letters, 28(1), 1992.
109. K.BYARD, 'Family of binary sequences, with usefitl
110. correlation properties'. Electronic Letters, Vol. 29, Npv., 1993.
111. A.GAVISH, ABRAHAM LAMPEL, 'On ternary coii^len»entary
'112. sequences', IEEE Trans. IT, Vol. 40, March, 1994.
113. BOZTAS, P.V. Kumar, 'Binary sequences with Gold like correlation but larger linear span', IEEE Trans. IT, Vol. 40, March, 1994.
114, OJEPA, E.J. TACCONI, 'Huffinan sequences with uniform time energy distribution'. Signal Proc, Vol. 37, No. 1, May, 1994.
,115. FAN, M. DARNELL and B. HONARY, 'New class of polyphase sequences with two 'Ij valued auto and cross-correlation function'. Electronic Letters, Vol. 30, June, 94.
128
Publications from Research Work :
13. Z.A. Abbasi and F. Ghani, "Multilevel Sequences with Good Autocorrelation Properties", Electronics Letters, ffiE; U.K., March 1988.
14. Z.A. Abbasi, "Iterative Method for Range Side Lobe Suppression for Binary Codes", Electronics Letters, lEE; U.K., July 1988.
15. Z.A. Abbasi and F. Ghani, "Range Side Lobe Elimination for Phase Coded Waveform", N.S.C., March 1990.
16. Z.A. Abbasi and F. Ghani, "Non-uniform Codes with Good Autocorrelation Properties", N.S.C., Jan. 1995, Agra.
17. Z.A. Abbasi, F. Ghani, Omar Farooq, "Range Side Lobe Suppression for Phase Coded Waveforms Using Neural Network", ICARCU '96, Westin Stanford, Singapore, Dec. 1996.